Journal articles: 'Derivative propagation method'

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Kim, Hyo Jin, Sang Ho Lee, and Moon Kyum Kim. "Prediction of Crack Propagation under Dynamic Loading Conditions by Using the Enhanced Point Collocation Meshfree Method." Key Engineering Materials 324-325 (November 2006): 1059–62. http://dx.doi.org/10.4028/www.scientific.net/kem.324-325.1059.

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An efficient and accurate numerical program with enhanced point collocation meshfree method is developed to simulate crack propagation under dynamic loading conditions. The enhanced meshfree method with point collocation formulation and derivative approximation in solids is presented. This study also presents the crack propagation criterion and computation of propagating direction, and the total structure of the numerical program named PCMDYC(Point Collocation Meshfree method for DYnamic Crack propagation). Several examples of crack propagation under dynamic loads are analyzed to simulate the arbitrary crack propagation under dynamic loads. The results show that PCMDYC predicts the propagating path of crack under dynamic loading conditions accurately and robustly.

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Trahan, Corey J., Robert E. Wyatt, and Bill Poirier. "Multidimensional quantum trajectories: Applications of the derivative propagation method." Journal of Chemical Physics 122, no. 16 (April 22, 2005): 164104. http://dx.doi.org/10.1063/1.1884606.

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Liao, Xuan, Tong Zhou, Longlong Zhang, Xiang Hu, and Yuanxi Peng. "A Method for Calculating the Derivative of Activation Functions Based on Piecewise Linear Approximation." Electronics 12, no. 2 (January 4, 2023): 267. http://dx.doi.org/10.3390/electronics12020267.

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Nonlinear functions are widely used as activation functions in artificial neural networks, which have a great impact on the fitting ability of artificial neural networks. Due to the complexity of the activation function, the computation of the activation function and its derivative requires a lot of computing resources and time during training. In order to improve the computational efficiency of the derivatives of the activation function in the back-propagation of artificial neural networks, this paper proposes a method based on piecewise linear approximation method to calculate the derivative of the activation function. This method is hardware-friendly and universal, it can efficiently compute various nonlinear activation functions in the field of neural network hardware accelerators. In this paper, we use least squares to improve a piecewise linear approximation calculation method that can control the absolute error and get less number of segments or smaller average error, which means fewer hardware resources are required. We use this method to perform a segmented linear approximation to the original or derivative function of the activation function. Both types of activation functions are substituted into a multilayer perceptron for binary classification experiments to verify the effectiveness of the proposed method. Experimental results show that the same or even slightly higher classification accuracy can be achieved by using this method, and the computation time of the back-propagation is reduced by 4–6% compared to the direct calculation of the derivative directly from the function expression using the operator encapsulated in PyTorch. This shows that the proposed method provides an efficient solution of nonlinear activation functions for hardware acceleration of neural networks.

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Irshad, Hajira, Mehnaz Shakeel, Imtiaz Ahmad, Hijaz Ahmad, Chutarat Tearnbucha, and Weerawat Sudsutad. "Simulation of generalized time fractional Gardner equation utilizing in plasma physics for non-linear propagation of ion-acoustic waves." Thermal Science 27, Spec. issue 1 (2023): 121–28. http://dx.doi.org/10.2298/tsci23s1121i.

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In this work, radial basis function collocation method (RBFCM) is implemented for generalized time fractional Gardner equation (GTFGE). The RBFCM is meshless and easy-to-implement in complex geometries and higher dimensions, therefore, it is highly demanding. In this work, the Caputo derivative of fractional order ? ? (0, 1] is used to approximate the first order time derivative whereas, Crank-Nicolson scheme is hired to approximate space derivatives. The numerical solutions are presented and discussed, which demonstrate that the method is effective and accurate.

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Rabie, Wafaa B., Hamdy M. Ahmed, Taher A. Nofal, and Soliman Alkhatib. "Wave solutions for the (3+1)-dimensional fractional Boussinesq-KP-type equation using the modified extended direct algebraic method." AIMS Mathematics 9, no. 11 (2024): 31882–97. http://dx.doi.org/10.3934/math.20241532.

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In this study, we introduce the new (3+1)-dimensional $ \beta $-fractional Boussinseq-Kadomtsev-Petviashvili (KP) equation that describes the wave propagation in fluid dynamics and other physical contexts. By using the modified extended direct algebraic method, we investigate diverse wave solutions for the proposed fractional model. The acquired solutions, include (dark, bright) soliton, hyperbolic, rational, exponential, Jacobi elliptic function, and Weierstrass elliptic doubly periodic solutions. The primary objective is to investigate the influence of fractional derivatives on the characteristics and dynamics of wave solutions. Graphical illustrations are presented to demonstrate the distinct changes in the amplitude, shape, and propagation patterns of the soliton solutions as the fractional derivative parameters are varied.

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Zeng, Huahui, Yanxiang Wang, Yang Zhou, Huijie Meng, Qigang Zhou, and Baozhong Jin. "Accurate Pseudo-Spectral Acoustic Wave Modelling with Time Dispersion Elimination." Applied Sciences 14, no. 19 (September 27, 2024): 8725. http://dx.doi.org/10.3390/app14198725.

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We propose an accurate method for modeling acoustic wave propagation. The spatial derivatives are calculated using Fourier transform to reduce spatial numerical dispersion. The standard staggered grid is adopted to suppress the non-causal ringing artifacts as in the traditional pseudo-spectral method. Moreover, to eliminate time dispersion arising from the discretization of the time derivative, an additional time-dispersion elimination term is introduced. As a result, the present method not only retains the advantages of the conventional pseudo-spectral method such as coarser spatial sampling or higher spatial derivative approximation accuracy, but also achieves higher temporal derivative approximation accuracy due to the adoption of the additional time-dispersion elimination term. Numerical examples demonstrate that the temporal dispersion elimination process can be contaminated by spatial numerical dispersion. Thus, the temporal and spatial numerical dispersions should be handled simultaneously, as proposed in this paper, to achieve accurate acoustic simulation.

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Soliman, Mahmoud, Hamdy M. Ahmed, Niveen Badra, Taher A. Nofal, and Islam Samir. "Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method." AIMS Mathematics 9, no. 9 (2024): 25205–22. http://dx.doi.org/10.3934/math.20241229.

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We investigated the dynamics of highly dispersive nonlinear gap solitons in optical fibers with dispersive reflectivity, utilizing a conformable fractional derivative model. The modified extended direct algebraic method was employed to obtain various soliton solutions, including bright solitons and singular solitons, as well as hyperbolic and trigonometric solutions. The key findings demonstrated that the fractional derivative parameter ($ \alpha $) can effectively control the wave propagation, causing a shift in the wave signal while maintaining the same amplitude. This is a novel contribution, as the ability to control soliton properties through the conformable derivative is explored for the first time in this work. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. This research provides insights into the potential applications of fractional calculus in the design and optimization of photonic devices for optical communication systems.

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Hsu, Yupai P. "Multilayer dielectric inversion for electromagnetic propagation logging." GEOPHYSICS 57, no. 10 (October 1992): 1260–69. http://dx.doi.org/10.1190/1.1443194.

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A new method for dielectric constant and conductivity inversion is described in this article. The method consists of a two‐step process. A mathematical operation named the ‘Pinched Second Derivative” is first defined and applied to identify the bed boundaries of a multilayered medium. The operation is based on the fact that when a receiver crosses a bed boundary, the received signal suffers a discontinuity in the second derivative in proportion to the dielectric contrast between the adjacent beds. An oscillating multilayer forward model is then used to parametrically invert the dielectric constant and conductivity of each layer of the medium. The inversion involves a two‐dimensional (2-D)‐moving‐grid search and an error function defined through a Chebyshev type metric.

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Xavier, Marcel, and Nicolas Van Goethem. "Brittle fracture on plates governed by topological derivatives." Engineering Computations 39, no. 1 (September 30, 2021): 421–37. http://dx.doi.org/10.1108/ec-07-2021-0375.

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PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.

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Izgec, B., and C. S. S. Kabir. "Identification and Characterization of High-Conductive Layers in Waterfloods." SPE Reservoir Evaluation & Engineering 14, no. 01 (December 20, 2010): 113–19. http://dx.doi.org/10.2118/123930-pa.

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Summary This study expands upon the use of modified-Hall analysis (MHA) to discern the characteristics of a high-permeability channel. Briefly, the modified-Hall plot uses three curves involving improved Hall-integral (H-I) and the two derivatives, analytic and numeric. Ordinarily, the derivative curves overlay on the integral curve during matrix injection, but separate lower when fracturing occurs. This work presents a method to identify and characterize high-conductive layers or channels between injector and producer pairs with the MHA. The distance separating the integral and derivative curves provides the required information to quantify channel properties. A simple analytical solution is presented for transforming the separation distance into channel permeabilitythickness product. The analytic derivative is based on the radial-flow-pattern assumption and the numeric derivative is correlated to the pressure response. Therefore, a comparison of these two curves reveals clues about the maturity of a waterflood at a given time. Several simulated examples verified the channel-property-estimation algorithm and identified the distinctive derivative signatures for channeling and fracturing situations. This method is also useful for identification of wormhole propagation during sand production in unconsolidated formations.

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Wang, Yanfei, Yaxin Ning, and Yibo Wang. "Fractional Time Derivative Seismic Wave Equation Modeling for Natural Gas Hydrate." Energies 13, no. 22 (November 12, 2020): 5901. http://dx.doi.org/10.3390/en13225901.

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Simulation of the seismic wave propagation in natural gas hydrate (NGH) is of great importance. To finely portray the propagation of seismic wave in NGH, attenuation properties of the earth’s medium which causes reduced amplitude and dispersion need to be considered. The traditional viscoacoustic wave equations described by integer-order derivatives can only nearly describe the seismic attenuation. Differently, the fractional time derivative seismic wave-equation, which was rigorously derived from the Kjartansson’s constant-Q model, could be used to accurately describe the attenuation behavior in realistic media. We propose a new fractional finite-difference method, which is more accurate and faster with the short memory length. Numerical experiments are performed to show the feasibility of the proposed simulation scheme for NGH, which will be useful for next stage of seismic imaging of NGH.

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Qiu, S., H. Liu, and WP Li. "Turbofan duct geometry optimization for low noise using remote continuous adjoint method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 229, no. 1 (April 24, 2014): 69–90. http://dx.doi.org/10.1177/0954406214532631.

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In this paper, a remote continuous adjoint-based acoustic propagation (RABAP) method is proposed for low noise turbofan duct design. The goal is to develop a set of adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the amplitude of sound pressure at a near-field location. The governing equations for the 2.5D acoustic perturbation equation solver (APE) formulation for duct acoustic propagation is first introduced. This is followed by the formulation and discretization of the remote continuous adjoint equations based on 2.5D APE. The special treatment of the adjoint boundary condition to obtain sensitivities derivatives is also discussed. The theory is applied to acoustic design of an axisymmetric fan bypass duct for two different tone noise radiations. The 2.5D APE is further validated using comparisons to an experiment data of the TURNEX nozzle geometry. The implementation of the remote continuous adjoint method is validated by comparing the sensitivity derivative with that obtained using finite difference method. The result obtained confirms the effectiveness and efficiency of the proposed RABAP framework.

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Bongiorno, Jacopo, and Andrea Mariscotti. "Uncertainty and Sensitivity of the Feature Selective Validation (FSV) Method." Electronics 11, no. 16 (August 13, 2022): 2532. http://dx.doi.org/10.3390/electronics11162532.

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The FSV method is a recognized validation tool that initially assesses the similarity between data sets for electromagnetic measurements and models. Its use may be extended to many problems and applications, and in particular, with relation to electrical systems, but it should be characterized in terms of its uncertainty, as for measurement tools. To this aim, the Guide to the Expression of Uncertainty in Measurement (GUM) is applied for the propagation of uncertainty from the experimental data to the Feature Selective Validation (FSV) quantities, using Monte Carlo analysis as confirmation, which ultimately remains the most reliable approach to determine the propagation of uncertainty, given the significant FSV non-linearity. Such non-linearity in fact compromises the accuracy of the Taylor approximation supporting the use of first-order derivatives (and derivative terms in general). MCM results are instead more stable and show sensitivity vs. input data uncertainty in the order of 10 to 100, highly depending on the local data samples value. To this aim, normalized sensitivity coefficients are also reported, in an attempt to attenuate the scale effects, redistributing the observed sensitivity values that, however, remain in the said range, up to about 100.

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Wang, Xiaoming, Rimsha Ansar, Muhammad Abbas, Farah Aini Abdullah, and Khadijah M. Abualnaja. "The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches." Axioms 12, no. 6 (June 16, 2023): 599. http://dx.doi.org/10.3390/axioms12060599.

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The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

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Shqair, Mohammed, Mohammed Alabedalhadi, Shrideh Al-Omari, and Mohammed Al-Smadi. "Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method." Fractal and Fractional 6, no. 5 (May 5, 2022): 252. http://dx.doi.org/10.3390/fractalfract6050252.

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The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

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Puchalski, Jacek. "Nonlinear Curve Fitting to Measurement Points with WTLS Method Using Approximation of Linear Model." International Journal of Automation, Artificial Intelligence and Machine Learning 4, no. 1 (June 28, 2024): 36–60. http://dx.doi.org/10.61797/ijaaiml.v4i1.326.

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The paper presents an approximate method of fitting measurement points to parameterized arbitrary nonlinear curves described by complex equations, even implicit ones, the most commonly used method of least squares in general WTLS. An approximation of a linear model is used here, in which the laws of propagation of error and propagation of uncertainty are true, so that only the first derivative of the transforming function is relevant. The effectiveness of the method has been demonstrated in several numerical examples. The method was verified on several nonlinear functions using the iterative algorithm by Monte Carlo propagation of distribution and the classical method based on the Levenberg-Marquardt algorithm for nonlinear optimization.

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Chu, Chunlei, and Paul L. Stoffa. "Implicit finite-difference simulations of seismic wave propagation." GEOPHYSICS 77, no. 2 (March 2012): T57—T67. http://dx.doi.org/10.1190/geo2011-0180.1.

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We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples.

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Ramos, J. I. "A conservative, spatially continuous method of lines for one-dimensional reaction-diffusion equations." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 11 (November 6, 2017): 2650–78. http://dx.doi.org/10.1108/hff-12-2016-0483.

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Purpose The purpose of this paper is to develop a new finite-volume method of lines for one-dimensional reaction-diffusion equations that provides piece-wise analytical solutions in space and is conservative, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficient on wave propagation. Design/methodology/approach A conservative, finite-volume method of lines based on piecewise integration of the diffusion operator that provides a globally continuous approximate solution and is second-order accurate is presented. Numerical experiments that assess the accuracy of the method and the time required to achieve steady state, and the effects of the nonlinear diffusion coefficients on wave propagation and boundary values are reported. Findings The finite-volume method of lines presented here involves the nodal values and their first-order time derivatives at three adjacent grid points, is linearly stable for a first-order accurate Euler’s backward discretization of the time derivative and has a smaller amplification factor than a second-order accurate three-point centered discretization of the second-order spatial derivative. For a system of two nonlinearly-coupled, one-dimensional reaction-diffusion equations, the amplitude, speed and separation of wave fronts are found to be strong functions of the dependence of the nonlinear diffusion coefficients on the concentration and temperature. Originality/value A new finite-volume method of lines for one-dimensional reaction-diffusion equations based on piecewise analytical integration of the diffusion operator and the continuity of the dependent variables and their fluxes at the cell boundaries is presented. The method may be used to study heat and mass transfer in layered media.

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Heaney, Kevin D. "Efficient parabolic equation based travel time computation." Journal of the Acoustical Society of America 154, no. 4_supplement (October 1, 2023): A83. http://dx.doi.org/10.1121/10.0022876.

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In this paper an efficient method for computing the travel time of an acoustic wave as a function of range using the parabolic equation model. The frequency derivative of the acoustic phase is the differential travel time associated with a propagation in range. By taking this difference across closely spaced frequencies (0.02 f0) and integrating in range, this method computes the travel time of the dominant acoustic arrival. The method compares well with other travel time methods for four different cases, including deep water (shallow source, axial source), upslope and shallow water, and a global 3D propagation environment.

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Gomez, J. F., and B. Ghanbari. "The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative." Revista Mexicana de Física 65, no. 5 Sept-Oct (September 2, 2019): 503. http://dx.doi.org/10.31349/revmexfis.65.503.

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In this paper, the generalized exponential rational function method (GERFM) and the extended sinh-Gordon equation expansion method (ShGEEM) are used to construct exact solutions of the perturbed β-conformable-time Radhakrishnan-Kundu-Lakshmanan (RKL) equation. This model governs soliton propagation dynamics through a polarization-preserving fiber. Fractional derivatives are described in the β-conformable sense. As a result, we get new form of solitary traveling wave solutions for this model including novel soliton, traveling waves and kink-type solutions with complex structures. Physical interpretations of some extracted solutions are also included through taking suitable values of parameters and derivative order in them. It is proved that these methods are powerful, efficient, and can be fruitfully implemented to establish new solutions of nonlinear conformable-time partial differential equations applied in mathematical physics.

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Aderyani, Safoura Rezaei, Reza Saadati, Donal O’Regan, and Fehaid Salem Alshammari. "Describing Water Wave Propagation Using the G′G2–Expansion Method." Mathematics 11, no. 1 (December 29, 2022): 191. http://dx.doi.org/10.3390/math11010191.

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In the present study, our focus is to obtain the different analytical solutions to the space–time fractional Bogoyavlenskii equation in the sense of the Jumaries-modified Riemann–Liouville derivative and to the conformable time–fractional-modified nonlinear Schrödinger equation that describes the fluctuation of sea waves and the propagation of water waves in ocean engineering, respectively. The G′G2–expansion method is applied to investigate the dynamics of solitons in relation to governing models. Moreover, the restriction conditions for the existence of solutions are reported. In addition, we note that the accomplished solutions are useful to the description of wave fluctuation and the wave propagation survey and are also significant for experimental and numerical verification in ocean engineering.

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Wenk, S., C. Pelties, H. Igel, and M. Käser. "Regional wave propagation using the discontinuous Galerkin method." Solid Earth Discussions 4, no. 2 (August 23, 2012): 1129–64. http://dx.doi.org/10.5194/sed-4-1129-2012.

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Abstract. We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. The ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper-mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

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Wenk, S., C. Pelties, H. Igel, and M. Käser. "Regional wave propagation using the discontinuous Galerkin method." Solid Earth 4, no. 1 (January 30, 2013): 43–57. http://dx.doi.org/10.5194/se-4-43-2013.

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Abstract. We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. This ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

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Li, Fengling, Zhixiang Hou, and Juan Chen. "A self-learning propotional–integral–derivative control of grouting pressure using the back-propagation model." Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 232, no. 8 (May 19, 2018): 1090–99. http://dx.doi.org/10.1177/0959651818774485.

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For the security of grouting process of dam foundation, grouting pressure control is one of the most important problems. In order to avoid dangerous grouting pressure fluctuation and improve the control precision, a feedback propotional–integral–derivative control method was presented for the whole grouting system. Because the grouting pressure is affected by many factors such as grouting flow, grouts density, and geological conditions, the parameters of propotional–integral–derivative must be tuned. In this article, the adaptive tuning method is presented. The back-propagation artificial neural networks model was proposed to simulate the grouting control process, and sensitivity analysis algorithm based on orthogonal test method was adopted for the selection of input variables. To obtain the optimal propotional–integral–derivative parameters, an iteration algorithm was used in each sampling interval time and the discrete Lyapunov function of the tracking error. The simulation results showed that self-learning propotional–integral–derivative tuning was robust and effective for the realization of the automatic control device in the grouting process.

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Tessmer, E., and D. Kosloff. "3-D elastic modeling with surface topography by a Chebychev spectral method." GEOPHYSICS 59, no. 3 (March 1994): 464–73. http://dx.doi.org/10.1190/1.1443608.

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The 3-D numerical Chebychev modeling scheme accounts for surface topography. The method is based on spectral derivative operators. Spatial differencing in horizontal directions is performed by the Fourier method, whereas vertical derivatives are carried out by a Chebychev method that allows for the incorporation of boundary conditions into the numerical scheme. The method is based on the velocity‐stress formulation. The implementation of surface topography is done by mapping a rectangular grid onto a curved grid. Boundary conditions are applied by means of characteristic variables. The study of surface effects of seismic wave propagation in the presence of surface topography is important, since nonray effects such as diffractions and scattering at rough surfaces must be considered. Several examples show this. The 3-D modeling alogrithm can serve as a tool for understanding these phenomena since it computes the full wavefield.

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Tessmer, Ekkehart. "Using the rapid expansion method for accurate time-stepping in modeling and reverse-time migration." GEOPHYSICS 76, no. 4 (July 2011): S177—S185. http://dx.doi.org/10.1190/1.3587217.

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Reverse-time migration is based on seismic forward modeling algorithms, where spatial derivatives usually are calculated by finite differences or by the Fourier method. Time integration in general is done by finite-difference time stepping of low orders. If the spatial derivatives are calculated by high-order methods and time stepping is based on low-order methods, there is an imbalance that might require that the time-step size needs to be very small to avoid numerical dispersion. As a result, computing times increase. Using the rapid expansion method (REM) avoids numerical dispersion if the number of expansion terms is chosen properly. Comparisons with analytical solutions show that the REM is preferable, especially at larger propagation times. For reverse-time migration, the REM needs to be applied in a time-stepping manner. This is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time. This is not appropriate for reverse-time migration where the sources have different time histories. In reverse-time migration, it might be desirable to use the Poynting vector information to estimate opening angles to improve the quality of the image. In the solution of the wave equation, this requires that one calculates not only the pressure wavefield but also its time derivative. The rapid expansion method can be extended easily to provide this time derivative with negligible extra cost.

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Akkas, N., and F. Erdogan. "The Residual Variable Method Applied to Acoustic Wave Propagation from a Spherical Surface." Journal of Vibration and Acoustics 115, no. 1 (January 1, 1993): 75–80. http://dx.doi.org/10.1115/1.2930318.

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The classical wave equation in spherical coordinates is expressed in terms of a residual potential applying the Residual Variable Method. This method essentially eliminates the second derivative of the potential with respect to the radial coordinate from the wave equation. Thus, the dynamic pressure distribution on the surface of a spherical cavity can be studied by considering the cavity surface only. Moreover, the Residual Variable Method, being amenable to “marching” solutions in a finite-difference implementation, is very suitable for the analysis of acoustic wave propagation into the finite medium from the cavity surface. The propagation of the wave from the internal surface can be followed numerically. There is no need to discretize the infinite domain in its entirety at all. The propagation analysis can be terminated at any point in the radial direction without having to consider the rest.

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Chou, Chia-Chun, and Robert E. Wyatt. "Trajectory approach to quantum wave packet dynamics: The correlated derivative propagation method." Chemical Physics Letters 500, no. 4-6 (November 2010): 342–46. http://dx.doi.org/10.1016/j.cplett.2010.10.039.

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Trahan, Corey J., Keith Hughes, and Robert E. Wyatt. "A new method for wave packet dynamics: Derivative propagation along quantum trajectories." Journal of Chemical Physics 118, no. 22 (June 8, 2003): 9911–14. http://dx.doi.org/10.1063/1.1578061.

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Mishra, Suchana, Rabindra Kishore Mishra, and Srikanta Patnaik. "Discrete (G'/G )-expansion: a Method Used to Get Exact Solution of Fdde (Fractional Differential-difference Equation) Linked With Nltl (Non-linear Transmission Line)." International Journal of Circuits, Systems and Signal Processing 15 (May 18, 2021): 453–60. http://dx.doi.org/10.46300/9106.2021.15.49.

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Here, we have used the discrete (G'/G)-expansion procedure with the derivative operator MR-L (modified Riemann-Liouville) and FCT (fractional complex transform) to find the exact/analytical solution of an electrical transmission line which is non-linear. Results include solutions for integer and fractional DDE. We consider two special cases of solutions: hyperbolic and trigonometric. Hyperbolic solutions indicate propagation of singular wave on the transmission line. Trigonometric solutions show propagation of complex wave.

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Christou, M. A., and I. C. Christov. "Christov expansion method for nonlocal nonlinear evolution equations." Journal of Physics: Conference Series 2675, no. 1 (December 1, 2023): 012022. http://dx.doi.org/10.1088/1742-6596/2675/1/012022.

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Abstract Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

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King, Matthew J., Timon S. Gutleb, Ben Cox, and Bradley Treeby. "A static memory method for modelling time-fractional power law absorption." Journal of the Acoustical Society of America 155, no. 3_Supplement (March 1, 2024): A290. http://dx.doi.org/10.1121/10.0027538.

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The attenuation of ultrasound propagation through tissue is known to follow a frequency power-law in the time domain. This can be modelled as a loss operator in the equation of state within the Euler equations that takes the form of a fraction time derivative operator. This can be treated through a second order accurate transfer between the fractional time derivative and a fractional Laplacian spacial operator in order to avoid storage of the full time history. This is used for example by the k-Wave toolbox. As an alternate finite history methods have been suggested. Building on recent work, here, we re-write the time fractional derivative as a finite sum using a recursion relation. This allows the time fractional derivative to be directly computed with a static memory requirement. For homogeneous media, the advantage of this over the fractional Laplacian methods may not initially be obvious with a base higher memory requirement and computational cost with only a small increase in accuracy. However, upon introducing a heterogeneous medium with regions of different power law attenuation; the increased computational cost of the time fractional method is contained exclusively within the pre-computation, while the increased cost for the fractional Laplacian is applied to each time step.

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Rajpoot, Manoj K., Vivek S. Yadav, Jyoti Jaglan, and Ankit Singh. "Sound and soliton wave propagation in homogeneous and heterogeneous mediums with the new two-derivative implicit–explicit Runge–Kutta–Nyström method." AIP Advances 12, no. 7 (July 1, 2022): 075110. http://dx.doi.org/10.1063/5.0099853.

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This paper derives a new family of implicit–explicit time-marching methods for PDEs with the second-order derivative in time. The present implicit method is based on the two-derivative Runge–Kutta–Nyström methods, which use a third-order time derivative of the solution. Although the current approach is implicit, it does not need to invert the coefficient matrix of the discretized system of equations. The stability properties are assessed using Fourier analysis for the model test problems by considering space–time discretizations together. The present methods are validated by comparing to some of the most widely used time-marching methods available in the literature. In addition, to assess the robustness and efficiency of the present methods, we have also performed numerical simulations of acoustic wave propagation in two- and three-layered heterogeneous media and sine-Gordon solitons for damped and undamped cases. Computed results match very well with the exact and numerical solutions noted in the literature.

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Liang, Xiao, and Bo Tang. "Efficient Exponential Time-Differencing Methods for the Optical Soliton Solutions to the Space-Time Fractional Coupled Nonlinear Schrödinger Equation." Journal of Mathematics 2021 (April 23, 2021): 1–10. http://dx.doi.org/10.1155/2021/5575128.

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The coupled nonlinear Schrödinger equation is used in simulating the propagation of the optical soliton in a birefringent fiber. Hereditary properties and memory of various materials can be depicted more precisely using the temporal fractional derivatives, and the anomalous dispersion or diffusion effects are better described by the spatial fractional derivatives. In this paper, one-step and two-step exponential time-differencing methods are proposed as time integrators to solve the space-time fractional coupled nonlinear Schrödinger equation numerically to obtain the optical soliton solutions. During this procedure, we take advantage of the global Padé approximation to evaluate the Mittag-Leffler function more efficiently. The approximation error of the Padé approximation is analyzed. A centered difference method is used for the discretization of the space-fractional derivative. Extensive numerical examples are provided to demonstrate the efficiency and effectiveness of the modified exponential time-differencing methods.

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WANG, JIANG, YINGJIE LIANG, LIN QIU, and XU YANG. "IMPROVED MACHINE LEARNING TECHNIQUE FOR SOLVING HAUSDORFF DERIVATIVE DIFFUSION EQUATIONS." Fractals 28, no. 04 (June 2020): 2050071. http://dx.doi.org/10.1142/s0218348x20500711.

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This study aims at combining the machine learning technique with the Hausdorff derivative to solve one-dimensional Hausdorff derivative diffusion equations. In the proposed artificial neural network method, the multilayer feed-forward neural network is chosen and improved by using the Hausdorff derivative to the activation function of hidden layers. A trial solution is a combination of the boundary and initial condition terms and the network output, which can approximate the analytical solution. To transform the original Hausdorff derivative equation into a minimization problem, an error function is defined, where the coefficients are approximated by using the gradient descent algorithm in the back-propagation process. Two numerical examples are given to illustrate the accuracy and the robustness of the proposed method. The obtained results show that the improved machine learning technique is efficient in computing the Hausdorff derivative diffusion equations both from computational accuracy and stability.

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Boje, Edward. "Representation of simulation errors in single step methods using state dependent noise." MATEC Web of Conferences 347 (2021): 00001. http://dx.doi.org/10.1051/matecconf/202134700001.

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The local error of single step methods is modelled as a function of the state derivative multiplied by bias and zero-mean white noise terms. The deterministic Taylor series expansion of the local error depends on the state derivative meaning that the local error magnitude is zero in steady state and grows with the rate of change of the state vector. The stochastic model of the local error may include a constant, “catch-all” noise term. A continuous time extension of the local error model is developed and this allows the original continuous time state differential equation to be represented by a combination of the simulation method and a stochastic term. This continuous time stochastic differential equation model can be used to study the propagation of the simulation error in Monte Carlo experiments, for step size control, or for propagating the mean and variance. This simulation error model can be embedded into continuous-discrete state estimation algorithms. Two illustrative examples are included to highlight the application of the approach.

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Odabasi Koprulu, Meryem, and Zehra Pinar Izgi. "Solitons of the Twin-Core Couplers with Fractional Beta Derivative Evolution in Optical Metamaterials via Two Distinct Methods." Journal of Mathematics 2024 (March 27, 2024): 1–14. http://dx.doi.org/10.1155/2024/8852337.

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The rapid advancements in metamaterial research have brought forth a new era of possibilities for controlling and manipulating light at the nanoscale. In particular, the design and engineering of optical metamaterials have created advances in the field of photonics, enabling the development of advanced devices with unprecedented functionalities. Among the myriad of intriguing metamaterial structures, the nonlinear directional couplers with beta derivative evolution have emerged as a significant avenue of exploration, offering remarkable potential for light propagation and manipulation. This study obtains the solitary wave solutions for twin-core couplers having spatial-temporal fractional beta derivative evolution by using two different methods, the Bernoulli method and the complete polynomial discriminant system method. By graphing some of the obtained solutions, the effect of the beta derivative has been shown. The findings would be beneficial to understand physical behaviours in nonlinear optics, particularly twin-core couplers with optical metamaterials.

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Jawad, Anwar Ja’afar Mohamad. "Three Different Methods for New Soliton Solutions of the Generalized NLS Equation." Abstract and Applied Analysis 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/5137946.

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Three different methods are applied to construct new types of solutions of nonlinear evolution equations. First, the Csch method is used to carry out the solutions; then the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions. The effectiveness of these methods is demonstrated by applications to the RKL model, the generalized derivative NLS equation. The solitary wave solutions and trigonometric function solutions are obtained. The obtained solutions are very useful in the nonlinear pulse propagation through optical fibers.

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Chen, Dan, and Zhao Li. "Traveling Wave Solution of the Kaup–Boussinesq System with Beta Derivative Arising from Water Waves." Discrete Dynamics in Nature and Society 2022 (December 6, 2022): 1–6. http://dx.doi.org/10.1155/2022/8857299.

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The main purpose of this paper is to construct the traveling wave solution of the Kaup–Boussinesq system with beta derivative arising from water waves. By using the complete discriminant system method of polynomial, the rational function solution, the trigonometric function solution, the exponential function solution, and the Jacobian function solution of the Kaup–Boussinesq system with beta derivative are obtained. In order to further explain the propagation of the Kaup–Boussinesq system with beta derivative in water waves, we draw its three-dimensional diagram, two-dimensional diagram, density plot, and contour plot by using Maple software.

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Hoffman, Adam J., and John C. Lee. "A time-dependent neutron transport method of characteristics formulation with time derivative propagation." Journal of Computational Physics 307 (February 2016): 696–714. http://dx.doi.org/10.1016/j.jcp.2015.10.039.

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Chou, Chia-Chun. "Complex-valued derivative propagation method with approximate Bohmian trajectories for quantum barrier scattering." Chemical Physics 457 (August 2015): 160–70. http://dx.doi.org/10.1016/j.chemphys.2015.06.008.

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Krause, Gustavo, Sergio Elaskar, and Andrea Costa. "Chaos and Intermittency in the DNLS Equation Describing the Parallel Alfvén Wave Propagation." Journal of Astrophysics 2014 (April 14, 2014): 1–15. http://dx.doi.org/10.1155/2014/812052.

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When the Hall effect is included in the magnetohydrodynamics equations (Hall-MHD model) the wave propagation modes become coupled, but for propagation parallel to the ambient magnetic field the Alfvén mode decouples from the magnetosonic ones, resulting in circularly polarized waves that are described by the derivative nonlinear Schrödinger (DNLS) equation. In this paper, the DNLS equation is numerically solved using spectral methods for the spatial derivatives and a fourth order Runge-Kutta scheme for time integration. Firstly, the nondiffusive DNLS equation is considered to test the validity of the method by verifying the analytical condition of modulational stability. Later, diffusive and excitatory effects are incorporated to compare the numerical results with those obtained by a three-wave truncation model. The results show that different types of attractors can exist depending on the diffusion level: for relatively large damping, there are fixed points for which the truncation model is a good approximation; for low damping, chaotic solutions appear and the three-wave truncation model fails due to the emergence of new nonnegligible modes.

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Vivas-Cortez, Miguel, Majeed A. Yousif, Pshtiwan Othman Mohammed, Alina Alb Lupas, Ibrahim S. Ibrahim, and Nejmeddine Chorfi. "Hyperbolic Non-Polynomial Spline Approach for Time-Fractional Coupled KdV Equations: A Computational Investigation." Symmetry 16, no. 12 (December 4, 2024): 1610. https://doi.org/10.3390/sym16121610.

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The time-fractional coupled Korteweg–De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable Hyperbolic Non-Polynomial Spline Method (CHNPSM), for solving TFCKdVEs. The method leverages the inherent symmetry in the structure of TFCKdVEs, exploiting conformable derivatives and hyperbolic non-polynomial spline functions to preserve the equations’ symmetry properties during computation. Additionally, first-derivative finite differences are incorporated to enhance the method’s computational accuracy. The convergence order, determined by studying truncation errors, illustrates the method’s conditional stability. To validate its performance, the CHNPSM is applied to two illustrative examples and compared with existing methods such as the meshless spectral method and Petrov–Galerkin method using error norms. The results underscore the CHNPSM’s superior accuracy, showcasing its potential for advancing numerical computations in the domain of TFCKdVEs and preserving essential symmetries in these physical systems.

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Qiu, S., WB Song, and H. Liu. "Shape optimization of a general bypass duct for tone noise reduction using continuous adjoint method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 1 (March 25, 2013): 119–34. http://dx.doi.org/10.1177/0954406213481915.

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A novel continuous adjoint-based acoustic propagation method is proposed for low-noise turbofan duct design. A fan bypass duct tonal noise propagation model that is verified by comparison with an analytical solution of the modal radiation from a semi-infinite duct with the shear layer is enhanced with its continuous adjoint formulation, having been applied to design the bypass duct. First, this article presents the complete formulation of the time-dependent optimal design problem. Second, a continuous adjoint-based acoustic propagation method for two-dimensional bypass duct configurations is derived and presented. This article aims at describing the potential of the adjoint technique for aeroacoustic shape optimization. The implementation of the unsteady aeroacoustic adjoint method is validated by comparing the sensitivity derivative with that obtained by finite differences. Using a continuous adjoint formulation, the necessary aerodynamic gradient information is obtained with large computational savings over traditional finite-difference methods. The examples presented demonstrate that the combination of a continuous-adjoint algorithm with a noise prediction method can be an efficient design tool in the bypass duct noise design problem.

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Ruhiat, Yayat, and Suherman Suherman. "Development of Heat Conduction Equation using a Heat Propagation Model on ERK Solar Dryer Plates." Physics Access 04, no. 01 (May 2024): 44–50. http://dx.doi.org/10.47514/phyaccess.2024.4.1.005.

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The heat conduction equation is a combination of first-order and second-order differential equations. Solving first-order differential equations is necessary to examine temperature as a function of time. Meanwhile, solving second-order differential equations is needed to examine temperature as a function of space. The heat flux equation is based on Fourier's law, which shows that temperature is a function of time and space. Understanding heat conduction can be improved by building a heat propagation model on the Solar ERK dryer plate. Analysis of heat propagation on the drying plate used the Finite Difference Approach (FDA) method with explicit and implicit schemes. With an explicit scheme, the FDA method calculates the temperature (T) at a point on the spatial derivative term, when T is at time t, while the implicit scheme calculates T at a point on the space derivative term when T is at time t+Δt. Heat propagation at each time change was analyzed by developing a program using the MATLAB 17 application. The results of the analysis show that there are differences in heat propagation between the explicit and implicit schemes. The convergence and stability of calculations in explicit schemes are unstable, causing problems at the time step. Meanwhile, the implicit scheme is carried out simultaneously on all nodes so that convergence and stability are easily maintained, and there are no time-step limitations.

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Massoun, Y., C. Cesarano, A. K. Alomari, and A. Said. "Numerical study of fractional phi-4 equation." AIMS Mathematics 9, no. 4 (2024): 8630–40. http://dx.doi.org/10.3934/math.2024418.

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<abstract>In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional derivatives, making it even more challenging. The homotopy analysis method can effectively handle these nonlinearities. Our objective was to illustrate the reliability and accuracy of our proposed algorithm, which we achieved through a comparative analysis against results obtained using the Yang transform decomposition method. Using the residual error to determine the optimal value of the convergence control parameter $ \hbar $, the results presented underscored the remarkable efficiency and accuracy of this approach.</abstract>

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Raza, Nauman, Saima Arshed, Kashif Ali Khan, and Dumitru Baleanu. "New and more fractional soliton solutions related to generalized Davey–Stewartson equation using oblique wave transformation." Modern Physics Letters B 35, no. 19 (May 4, 2021): 2150317. http://dx.doi.org/10.1142/s0217984921503176.

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The generalized fractional Davey–Stewartson (DSS) equation with fractional temporal derivative, which is used to explore the trends of wave propagation in water of finite depth under the effects of gravity force and surface tension, is considered in this paper. The paper addresses the full nonlinearity of the proposed model. To extract the oblique soliton solutions of the generalized fractional DSS (FDSS) equation is the dominant feature of this research. The conformable fractional derivative is used for fractional temporal derivative and oblique wave transformation is used for converting the proposed model into ordinary differential equation. Two state-of-the-art integration schemes, modified auxiliary equation (MAE) and generalized projective Riccati equations (GPREs) method have been employed for obtaining the desired oblique soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, and periodic solitary wave solutions. The occurrence of these results ensured by the limitations utilized is also exceptionally promising to additionally investigate the propagation of waves of finite depth. The latest found solutions with their existence criteria are considered. The 2D and 3D portraits are also shown for some of the reported solutions. From the graphical representations, it have been illustrated that the descriptions of waves are changed along with the change in fractional and obliqueness parameters.

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Hobiny, Aatef, and Ibrahim Abbas. "The Effect of a Nonlocal Thermoelastic Model on a Thermoelastic Material under Fractional Time Derivatives." Fractal and Fractional 6, no. 11 (November 2, 2022): 639. http://dx.doi.org/10.3390/fractalfract6110639.

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This article develops a novel nonlocal theory of generalized thermoelastic material based on fractional time derivatives and Eringen’s nonlocal thermoelasticity. An ultra-short pulse laser heats the surface of the medium’s surrounding plane. Using the Laplace transform method, the basic equations and their accompanying boundary conditions were numerically solved. The distribution of thermal stress, temperature and displacement are physical variables for which the eigenvalues approach was employed to generate the analytical solution. Visual representations were used to examine the influence of the nonlocal parameters and fractional time derivative parameters on the wave propagation distributions of the physical fields for materials. The consideration of the nonlocal thermoelasticity theory (nonlocal elasticity and heat conduction) with fractional time derivatives may lead us to conclude that the variations in physical quantities are considerably impacted.

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Aguilar, J. F. Gómez, T. Córdova-Fraga, J. Tórres-Jiménez, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, and G. V. Guerrero-Ramírez. "Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation." Mathematical Problems in Engineering 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/7845874.

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The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

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Khater, Mostafa, Raghda Attia, and Dianchen Lu. "Modified Auxiliary Equation Method versus Three Nonlinear Fractional Biological Models in Present Explicit Wave Solutions." Mathematical and Computational Applications 24, no. 1 (December 20, 2018): 1. http://dx.doi.org/10.3390/mca24010001.

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In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent the population density occurring as a result of population supply, a lengthy wave propagating in the positive x-direction, and the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes, respectively. We discuss these models in nonlinear fractional partial differential equation formulas. We used the conformable derivative properties to convert them into nonlinear ordinary differential equations with integer order. After adapting, we applied our new modification to these models to obtain solitary solutions of them. We obtained many novel solutions of these models, which serve to understand more about their properties. All obtained solutions were verified by putting them back into the original equations via computer software such as Maple, Mathematica, and Matlab.

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Journal articles on the topic 'Derivative propagation method'

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