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Journal articles on the topic 'Partial Numerical solutions'

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Published: 4 June 2021

Last updated: 24 January 2023

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1

Zhang, Zhao. "Numerical Analysis and Comparison of Gridless Partial Differential Equations." International Journal of Circuits, Systems and Signal Processing 15 (August 31, 2021): 1223–31. http://dx.doi.org/10.46300/9106.2021.15.133.

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In the field of science and engineering, partial differential equations play an important role in the process of transforming physical phenomena into mathematical models. Therefore, it is very important to get a numerical solution with high accuracy. In solving linear partial differential equations, meshless solution is a very important method. Based on this, we propose the numerical solution analysis and comparison of meshless partial differential equations (PDEs). It is found that the interaction between the numerical solutions of gridless PDEs is better, and the absolute error and relative error are lower, which proves the superiority of the numerical solutions of gridless PDEs

2

Wu, G., Eric Wai Ming Lee, and Gao Li. "Numerical solutions of the reaction-diffusion equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 2 (March 2, 2015): 265–71. http://dx.doi.org/10.1108/hff-04-2014-0113.

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Purpose – The purpose of this paper is to introduce variational iteration method (VIM) to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations. The Lagrange multipliers become the integral kernels. Design/methodology/approach – Using the discrete numerical integral formula, the general way is given to solve the famous reaction-diffusion equation numerically. Findings – With the given explicit solution, the results show the conveniences of the general numerical schemes and numerical simulation of the reaction-diffusion is finally presented in the cases of various coefficients. Originality/value – The method avoids the treatment of the time derivative as that in the classical finite difference method and the VIM is introduced to construct equivalent integral equations for initial-boundary value problems of nonlinear partial differential equations.

3

Wang, Zhigang, Xiaoting Liu, Lijun Su, and Baoyan Fang. "Numerical Solutions of Convective Diffusion Equations using Wavelet Collocation Method." Advances in Engineering Technology Research 1, no. 1 (May 17, 2022): 192. http://dx.doi.org/10.56028/aetr.1.1.192.

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Some partial differential equations appear in many application fields. Therefore, the discussion of numerical solutions of those partial differential equations using numerical methods becomes a valuable and important issue in numerical simulation. In numerical methods, the wavelet-collocation method has been frequently developed for solving PDEs, and the algorithm has yielded substantial results. However, theoretical research of the numerical solution has been rarely discussed yet. In this paper, the numerical solution of convective diffusion equations using the wavelet-collocation method is established, and its existence and uniqueness are derived.

4

NAKAO, Mitsuhiro. "Numerical Verification of Solutions for Partial Differential Equations." IEICE ESS FUNDAMENTALS REVIEW 2, no. 3 (2009): 19–28. http://dx.doi.org/10.1587/essfr.2.3_19.

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Nakao, Mitsuhiro T. "Numerical verification for solutions to partial differential equations." Sugaku Expositions 30, no. 1 (March 17, 2017): 89–109. http://dx.doi.org/10.1090/suga/419.

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6

Higdon, Robert L. "Numerical modelling of ocean circulation." Acta Numerica 15 (May 2006): 385–470. http://dx.doi.org/10.1017/s0962492906250013.

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Computational simulations of ocean circulation rely on the numerical solution of partial differential equations of fluid dynamics, as applied to a relatively thin layer of stratified fluid on a rotating globe. This paper describes some of the physical and mathematical properties of the solutions being sought, some of the issues that are encountered when the governing equations are solved numerically, and some of the numerical methods that are being used in this area.

7

Seth, G. S., S. Sarkar, and R. Sharma. "Effects of Hall current on unsteady hydromagnetic free convection flow past an impulsively moving vertical plate with Newtonian heating." International Journal of Applied Mechanics and Engineering 21, no. 1 (February 1, 2016): 187–203. http://dx.doi.org/10.1515/ijame-2016-0012.

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Abstract An investigation of unsteady hydromagnetic free convection flow of a viscous, incompressible and electrically conducting fluid past an impulsively moving vertical plate with Newtonian surface heating embedded in a porous medium taking into account the effects of Hall current is carried out. The governing partial differential equations are first subjected to the Laplace transformation and then inverted numerically using INVLAP routine of Matlab. The governing partial differential equations are also solved numerically by the Crank-Nicolson implicit finite difference scheme and a comparison has been provided between the two solutions. The numerical solutions for velocity and temperature are plotted graphically whereas the numerical results of skin friction and the Nusselt number are presented in tabular form for various parameters of interest. The present solution in special case is compared with a previously obtained solution and is found to be in excellent agreement.

8

Iqbal, Mazhar, M. T. Mustafa, and Azad A. Siddiqui. "A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/105414.

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Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.

9

Zou, Guang-an. "Numerical solutions to time-fractional stochastic partial differential equations." Numerical Algorithms 82, no. 2 (November 5, 2018): 553–71. http://dx.doi.org/10.1007/s11075-018-0613-0.

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10

ARLUKOWICZ, P., and W. CZERNOUS. "A numerical method of bicharacteristics For quasi-linear partial functional Differential equations." Computational Methods in Applied Mathematics 8, no. 1 (2008): 21–38. http://dx.doi.org/10.2478/cmam-2008-0002.

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Abstract Classical solutions of mixed problems for first order partial functional differential equations in several independent variables are approximated by solutions of an Euler-type difference problem. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that the given functions satisfy the nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from the general model by specializing the given operators.

11

Abhyankar, N. S., E. K. Hall, and S. V. Hanagud. "Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations." Journal of Applied Mechanics 60, no. 1 (March 1, 1993): 167–74. http://dx.doi.org/10.1115/1.2900741.

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The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

12

Kolokoltsov, Vassili, Feng Lin, and Aleksandar Mijatović. "Monte carlo estimation of the solution of fractional partial differential equations." Fractional Calculus and Applied Analysis 24, no. 1 (January 29, 2021): 278–306. http://dx.doi.org/10.1515/fca-2021-0012.

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Abstract The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

13

Kravchenko, Vladislav V., Josafath A. Otero, and Sergii M. Torba. "Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients." Advances in Mathematical Physics 2017 (2017): 1–5. http://dx.doi.org/10.1155/2017/2947275.

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A complete family of solutions for the one-dimensional reaction-diffusion equation, uxx(x,t)-q(x)u(x,t)=ut(x,t), with a coefficient q depending on x is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, the Dirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.

14

CHEN, C. S., SUNGWOOK LEE, and C. S. HUANG. "DERIVATION OF PARTICULAR SOLUTIONS USING CHEBYSHEV POLYNOMIAL BASED FUNCTIONS." International Journal of Computational Methods 04, no. 01 (March 2007): 15–32. http://dx.doi.org/10.1142/s0219876207001096.

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In this paper, we propose a simple and direct numerical procedure to obtain particular solutions for various types of differential equations. This procedure employs the power series expansion of a differential operator. Chebyshev polynomials are selected as basis functions for the approximation of the inhomogeneous terms of the given partial differential equation. This numerical scheme provides a highly efficient and accurate approximation for the evaluation of a particular solution for a variety of classes of partial differential equations. To demonstrate the effectiveness of the proposed scheme, we couple the method of fundamental solutions to solve a modified Helmholtz equation with irregular boundary configuration. The solutions were observed to have high accuracy.

15

Merdan, Mehmet, Ahmet Gökdoğan, Ahmet Yıldırım, and Syed Tauseef Mohyud-Din. "Numerical Simulation of Fractional Fornberg-Whitham Equation by Differential Transformation Method." Abstract and Applied Analysis 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/965367.

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An approximate analytical solution of fractional Fornberg-Whitham equation was obtained with the help of the two-dimensional differential transformation method (DTM). It is indicated that the solutions obtained by the two-dimensional DTM are reliable and present an effective method for strongly nonlinear partial equations. Exact solutions can also be obtained from the known forms of the series solutions.

16

Du, Mingjing. "An Improved Approach for Solving Partial Differential Equation Based on Reproducing Kernel Method." Security and Communication Networks 2021 (November 30, 2021): 1–5. http://dx.doi.org/10.1155/2021/7741274.

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The traditional reproducing kernel method (TRKM) cannot obtain satisfactory numerical results for solving the partial differential equation (PDE). In this study, for the first time, the abovementioned problems are solved by adaptive piecewise interpolation reproducing kernel method (APIRKM) to obtain the exact and approximate solutions of partial differential equations by means of series expansion using reconstructed kernel function. The highlight of this paper is to obtain more accurate approximate solution and save more time through adaptive discovery. Numerical solutions of the three examples show that the present method is more advantageous than TRKM.

17

Nur Aisyah Azeman, Siti, and . "Dual Solutions in the Boundary Layer Flow and Heat Transfer in the Presence of Thermal Radiation with Suction Effect." International Journal of Engineering & Technology 7, no. 4.33 (December 9, 2018): 17. http://dx.doi.org/10.14419/ijet.v7i4.33.23475.

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The dual solutions in the boundary layer flow and heat transfer in the presence of thermal radiation is quantitatively studied. The governing partial differential equations are derived into a system of ordinary differential equations using a similarity transformation, and afterward numerical solution obtained by a shooting technique. Dual solutions execute within a certain range of opposing and assisting flow which related to these numerical solutions. The similarity equations have two branches, upper or lower branch solutions, within a certain range of the mixed convection parameters. Further numerical results exist in our observations which enable to discuss the features of the respective solutions.

18

Piqueras, M. A., R. Company, and L. Jódar. "Stable Numerical Solutions Preserving Qualitative Properties of Nonlocal Biological Dynamic Problems." Abstract and Applied Analysis 2019 (July 1, 2019): 1–7. http://dx.doi.org/10.1155/2019/5787329.

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This paper deals with solving numerically partial integrodifferential equations appearing in biological dynamics models when nonlocal interaction phenomenon is considered. An explicit finite difference scheme is proposed to get a numerical solution preserving qualitative properties of the solution. Gauss quadrature rules are used for the computation of the integral part of the equation taking advantage of its accuracy and low computational cost. Numerical analysis including consistency, stability, and positivity is included as well as numerical examples illustrating the efficiency of the proposed method.

19

Secer, Aydin. "Sinc-Galerkin method for solving hyperbolic partial differential equations." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 8, no. 2 (July 24, 2018): 250–58. http://dx.doi.org/10.11121/ijocta.01.2018.00608.

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In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

20

Ahmad, Hijaz, Ali Akgül, Tufail A. Khan, Predrag S. Stanimirović, and Yu-Ming Chu. "New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations." Complexity 2020 (October 6, 2020): 1–10. http://dx.doi.org/10.1155/2020/8829017.

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The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.

21

Khatri Ghimire, B., H. Y. Tian, and A. R. Lamichhane. "Numerical solutions of elliptic partial differential equations using Chebyshev polynomials." Computers & Mathematics with Applications 72, no. 4 (August 2016): 1042–54. http://dx.doi.org/10.1016/j.camwa.2016.06.012.

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22

Jódar, L., and M. C. Casabán. "Convergent discrete numerical solutions of coupled mixed partial differential systems." Mathematical and Computer Modelling 34, no. 3-4 (August 2001): 283–97. http://dx.doi.org/10.1016/s0895-7177(01)00061-9.

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23

Ali, Ramzan, Azeem Shahzad, Masood Khan, and Muhammad Ayub. "Analytic and numerical solutions for axisymmetric flow with partial slip." Engineering with Computers 32, no. 1 (June 13, 2015): 149–54. http://dx.doi.org/10.1007/s00366-015-0405-2.

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24

Liu, Nanshan, and En-Bing Lin. "Legendre wavelet method for numerical solutions of partial differential equations." Numerical Methods for Partial Differential Equations 26, no. 1 (January 2010): 81–94. http://dx.doi.org/10.1002/num.20417.

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25

Sanz-Serna, J. M. "Symplectic integrators for Hamiltonian problems: an overview." Acta Numerica 1 (January 1992): 243–86. http://dx.doi.org/10.1017/s0962492900002282.

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In the sciences, situations where dissipation is not significant may invariably be modelled by Hamiltonian systems of ordinary, or partial, differential equations. Symplectic integrators are numerical methods specifically aimed at advancing in time the solution of Hamiltonian systems. Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness.

26

Yao, Jun, Wenchao Liu, and Zhangxin Chen. "Numerical Solution of a Moving Boundary Problem of One-Dimensional Flow in Semi-Infinite Long Porous Media with Threshold Pressure Gradient." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/384246.

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A numerical method is presented for the solution of a moving boundary problem of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient (TPG) for the case of a constant flow rate at the inner boundary. In order to overcome the difficulty in the space discretization of the transient flow region with a moving boundary in the process of numerical solution, the system of partial differential equations for the moving boundary problem is first transformed equivalently into a closed system of partial differential equations with fixed boundary conditions by a spatial coordinate transformation method. Then a stable, fully implicit finite difference method is adopted to obtain its numerical solution. Finally, numerical results of transient distance of the moving boundary, transient production pressure of wellbore, and formation pressure distribution are compared graphically with those from a published exact analytical solution under different values of dimensionless TPG as calculated from actual experimental data. Comparison analysis shows that numerical solutions are in good agreement with the exact analytical solutions, and there is a big difference of model solutions between Darcy's flow and the fluid flow in porous media with TPG, especially for the case of a large dimensionless TPG.

27

Al-Smadi, Mohammed, Asad Freihat, Hammad Khalil, Shaher Momani, and Rahmat Ali Khan. "Numerical Multistep Approach for Solving Fractional Partial Differential Equations." International Journal of Computational Methods 14, no. 03 (April 13, 2017): 1750029. http://dx.doi.org/10.1142/s0219876217500293.

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In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense, while the behavior of solutions for different values of fractional order [Formula: see text] compared with exact solutions is shown graphically. The analysis is accompanied by four test examples to demonstrate that the proposed approach is reliable, fully compatible with the complexity of these equations, and can be strongly employed for many other nonlinear problems in fractional calculus.

28

Thapa, Dhak Bahadur, and Kedar Nath Uprety. "Analytic and Numerical Solutions of Couette Flow Problem: A Comparative Study." Journal of the Institute of Engineering 12, no. 1 (March 6, 2017): 105–13. http://dx.doi.org/10.3126/jie.v12i1.16731.

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In this work, an incompressible viscous Couette flow is derived by simplifying the Navier-Stokes equations and the resulting one dimensional linear parabolic partial differential equation is solved numerically employing a second order finit difference Crank-Nicolson scheme. The numerical solution and the exact solution are presented graphically.Journal of the Institute of Engineering, 2016, 12(1): 105-113

29

Aljarrah, Hussam, Mohammad Alaroud, Anuar Ishak, and Maslina Darus. "Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations." Mathematics 9, no. 22 (November 11, 2021): 2868. http://dx.doi.org/10.3390/math9222868.

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In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

30

Suárez, Pablo U., and J. Héctor Morales. "Numerical Solutions of Two-Way Propagation of Nonlinear Dispersive Waves Using Radial Basis Functions." International Journal of Partial Differential Equations 2014 (August 3, 2014): 1–8. http://dx.doi.org/10.1155/2014/407387.

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We obtain the numerical solution of a Boussinesq system for two-way propagation of nonlinear dispersive waves by using the meshless method, based on collocation with radial basis functions. The system of nonlinear partial differential equation is discretized in space by approximating the solution using radial basis functions. The discretization leads to a system of coupled nonlinear ordinary differential equations. The equations are then solved by using the fourth-order Runge-Kutta method. A stability analysis is provided and then the accuracy of method is tested by comparing it with the exact solitary solutions of the Boussinesq system. In addition, the conserved quantities are calculated numerically and compared to an exact solution. The numerical results show excellent agreement with the analytical solution and the calculated conserved quantities.

31

Jacobs, B. A., and C. Harley. "Two Hybrid Methods for Solving Two-Dimensional Linear Time-Fractional Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/757204.

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A computationally efficient hybridization of the Laplace transform with two spatial discretization techniques is investigated for numerical solutions of time-fractional linear partial differential equations in two space variables. The Chebyshev collocation method is compared with the standard finite difference spatial discretization and the absolute error is obtained for several test problems. Accurate numerical solutions are achieved in the Chebyshev collocation method subject to both Dirichlet and Neumann boundary conditions. The solution obtained by these hybrid methods allows for the evaluation at any point in time without the need for time-marching to a particular point in time.

32

Liu, Yong Qing, Rong Jun Cheng, and Hong Xia Ge. "Element-Free Galerkin (EFG) Method for Time Fractional Partial Differential Equations." Applied Mechanics and Materials 101-102 (September 2011): 343–47. http://dx.doi.org/10.4028/www.scientific.net/amm.101-102.343.

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In this paper, the first order time derivative of time fractional partial differential equations are replaced by the Caputo fractional order derivative. We derive the numerical solution of this equation using the Element-free Galerkin (EFG) method. In order to obtain the discrete equation, a various method is used and the essential boundary conditions are enforced by the penalty method. Numerical examples are presented and the results are in good agreement with exact solutions.

33

FALCONE, M. "NUMERICAL METHODS FOR DIFFERENTIAL GAMES BASED ON PARTIAL DIFFERENTIAL EQUATIONS." International Game Theory Review 08, no. 02 (June 2006): 231–72. http://dx.doi.org/10.1142/s0219198906000886.

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In this paper we present some numerical methods for the solution of two-persons zero-sum deterministic differential games. The methods are based on the dynamic programming approach. We first solve the Isaacs equation associated to the game to get an approximate value function and then we use it to reconstruct approximate optimal feedback controls and optimal trajectories. The approximation schemes also have an interesting control interpretation since the time-discrete scheme stems from a dynamic programming principle for the associated discrete time dynamical system. The general framework for convergence results to the value function is the theory of viscosity solutions. Numerical experiments are presented solving some classical pursuit-evasion games.

34

Jeong, Darae, Seungsuk Seo, Hyeongseok Hwang, Dongsun Lee, Yongho Choi, and Junseok Kim. "Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/359028.

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We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. We use a finite difference method to numerically solve the equation. To show the efficiency of the given boundary condition, several numerical examples are presented. In numerical test, we investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error. Numerical results show that linear boundary condition is accurate and efficient among the other boundary conditions.

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SOLTANALIZADEH, Babak, Hamidreza ESMALIFALAK, Rasoul HEKMATI, Zahra SARMAST, and Sepideh SHABANI. "Numerical Analysis of the One-Demential Wave Equation Subject to a Boundary Integral Specification." Walailak Journal of Science and Technology (WJST) 15, no. 6 (August 28, 2016): 421–37. http://dx.doi.org/10.48048/wjst.2018.1153.

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In this paper a numerical technique is developed for the one-dimensional wave equation that combines classical and integral boundary conditions. A new matrix formulation technique with arbitrary polynomial bases is proposed for the analytical solution of this kind of partial differential equation. Not only have the exact solutions been achieved by the known forms of the series solutions, but also, for the finite terms of series, the corresponding numerical approximations have been computed. We give a simple and efficient algorithm based on an iterative process for numerical solution of the method. Some numerical examples are included to demonstrate the validity and applicability of the technique.

36

Menshikh, V. V., and N. E. Chirkova. "A Numerical Method for Optimization of the Arrangement of Elements of a Video Surveillance System Taking into Account their Safety." Vestnik Tambovskogo gosudarstvennogo tehnicheskogo universiteta 27, no. 1 (2021): 073–80. http://dx.doi.org/10.17277/vestnik.2021.01.pp.073-080.

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The article deals with the development of a numerical method for optimizing the arrangement of elements of a video surveillance system, taking into account their own safety. The necessity of using the branch and bound scheme in the development of this method is substantiated, which makes it possible to search for an optimal solution with high efficiency. Methods of forming a partial decision tree, evaluating partial solutions, traversing the vertices of a partial decision tree are determined. A numerical example of the implementation of the proposed method is shown.

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Gul, Haji, Sajjad Ali, Kamal Shah, Shakoor Muhammad, Thanin Sitthiwirattham, and Saowaluck Chasreechai. "Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation." Symmetry 13, no. 11 (November 19, 2021): 2215. http://dx.doi.org/10.3390/sym13112215.

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In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

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Mungkasi, Sudi, Andreas Prasetyadi, and F. A. R. Sambada. "Numerical Solutions to Fast Transient Pipe Flow Problems." Advanced Materials Research 1123 (August 2015): 27–30. http://dx.doi.org/10.4028/www.scientific.net/amr.1123.27.

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We promote a finite volume method to solve a water hammer problem numerically. This problem is of the type of fast transient pipe flow. The mathematical model governing the problem is a system of two simultaneous partial differential equations. As the system is hyperbolic, our choice of numerical method is appropriate. In particular, we consider water flows through a pipe from a pressurized water tank at one end to a valve at the other end. We want to know the pressure and velocity profile in the pipe when the valve closes as a function of time. We find that the finite volume method is very robust to solve the problem.

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Patrício, Fernanda Simões, Miguel Patrício, and Higinio Ramos. "Extrapolating for attaining high precision solutions for fractional partial differential equations." Fractional Calculus and Applied Analysis 21, no. 6 (December 19, 2018): 1506–23. http://dx.doi.org/10.1515/fca-2018-0079.

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Abstract This paper aims at obtaining a high precision numerical approximation for fractional partial differential equations. This is achieved through appropriate discretizations: firstly we consider the application of shifted Legendre or Chebyshev polynomials to get a spatial discretization, followed by a temporal discretization where we use the Implicit Euler method (although any other temporal integrator could be used). Finally, the use of an extrapolation technique is considered for improving the numerical results. In this way a very accurate solution is obtained. An algorithm is presented, and numerical results are shown to demonstrate the validity of the present technique.

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Yazıcı, Muhammet, and Harun Selvitopi. "Numerical methods for the multiplicative partial differential equations." Open Mathematics 15, no. 1 (November 22, 2017): 1344–50. http://dx.doi.org/10.1515/math-2017-0113.

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Abstract We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

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Choudhury, A. H. "Wavelet Method for Numerical Solution of Parabolic Equations." Journal of Computational Engineering 2014 (February 27, 2014): 1–12. http://dx.doi.org/10.1155/2014/346731.

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We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.

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Fakharany, M., R. Company, and L. Jódar. "Unconditional Positive Stable Numerical Solution of Partial Integrodifferential Option Pricing Problems." Journal of Applied Mathematics 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/960728.

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This paper is concerned with the numerical solution of partial integrodifferential equation for option pricing models under a tempered stable process known as CGMY model. A double discretization finite difference scheme is used for the treatment of the unbounded nonlocal integral term. We also introduce in the scheme the Patankar-trick to guarantee unconditional nonnegative numerical solutions. Integration formula of open type is used in order to improve the accuracy of the approximation of the integral part. Stability and consistency are also studied. Illustrative examples are included.

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Filobello-Nino, U., H. Vazquez-Leal, A. Sarmiento-Reyes, B. Benhammouda, V. M. Jimenez-Fernandez, D. Pereyra-Diaz, A. Perez-Sesma, et al. "Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip." International Scholarly Research Notices 2014 (November 24, 2014): 1–10. http://dx.doi.org/10.1155/2014/747098.

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The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient.

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Iqbal, Sajad, Mohammed K. A. Kaabar, and Francisco Martínez. "A Novel Homotopy Perturbation Algorithm Using Laplace Transform for Conformable Partial Differential Equations." Mathematical Problems in Engineering 2021 (December 14, 2021): 1–13. http://dx.doi.org/10.1155/2021/2573067.

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In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.

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Flyer, N., and A. S. Fokas. "A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2095 (April 2008): 1823–49. http://dx.doi.org/10.1098/rspa.2008.0041.

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A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

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Alesemi, Meshari, Jameelah S. Al Shahrani, Naveed Iqbal, Rasool Shah, and Kamsing Nonlaopon. "Analysis and Numerical Simulation of System of Fractional Partial Differential Equations with Non-Singular Kernel Operators." Symmetry 15, no. 1 (January 13, 2023): 233. http://dx.doi.org/10.3390/sym15010233.

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The exact solution to fractional-order partial differential equations is usually quite difficult to achieve. Semi-analytical or numerical methods are thought to be suitable options for dealing with such complex problems. To elaborate on this concept, we used the decomposition method along with natural transformation to discover the solution to a system of fractional-order partial differential equations. Using certain examples, the efficacy of the proposed technique is demonstrated. The exact and approximate solutions were shown to be in close contact in the graphical representation of the obtained results. We also examine whether the proposed method can achieve a quick convergence with a minimal number of calculations. The present approaches are also used to calculate solutions in various fractional orders. It has been proven that fractional-order solutions converge to integer-order solutions to problems. The current technique can be modified for various fractional-order problems due to its simple and straightforward implementation.

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Dey, Debasish, and Rupjyoti Borah. "DUAL SOLUTIONS OF BOUNDARY LAYER FLOW WITH HEAT AND MASS TRANSFERS OVER AN EXPONENTIALLY SHRINKING CYLINDER: STABILITY ANALYSIS." Latin American Applied Research - An international journal 50, no. 4 (June 24, 2020): 247–53. http://dx.doi.org/10.52292/j.laar.2020.535.

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Boundary layer flow with heat and mass transfers over a stretching/shrinking cylinder has been investigated. The governing partial differential equations are converted into a set of ordinary differential equations using suitable similarity transformations and have been solved numerically using MATLAB built in bvp4c solver technique. The numerical results are graphically discussed in the form of velocity, temperature and concentration distributions for various values of flow parameters. Numerical results show that dual solutions are possible in specific range of the suction parameter. A stability analysis is executed to obtain which solution is linearly stable and physically realizable.

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CAI, WEI, and FAJIE WANG. "NUMERICAL INVESTIGATION OF THREE-DIMENSIONAL HAUSDORFF DERIVATIVE ANOMALOUS DIFFUSION MODEL." Fractals 28, no. 02 (March 2020): 2050020. http://dx.doi.org/10.1142/s0218348x20500206.

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The Hausdorff derivative has been recognized as an efficient tool to characterize anomalous diffusion phenomena. This paper makes the first attempt to numerically investigate three-dimensional Hausdorff derivative diffusion equation by the method of fundamental solutions. The fundamental solution of the three-dimensional Hausdorff derivative diffusion equation is closely related to scaling transform and non-Euclidean Hausdorff fractal distance. The used method, as a meshless technique, is simple, accurate and efficient for solving the partial differential equations with fundamental solutions. Three numerical experiments have been conducted to reveal the effectiveness and rationality of Hausdorff derivative anomalous diffusion models with various temporal and spatial fractal orders, as well as the accuracy of the developed methodology.

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Jia, Yunfeng. "Analysis and simulation on dynamics of a partial differential system with nonlinear functional responses." Nonlinear Analysis: Modelling and Control 26, no. 2 (March 1, 2021): 293–314. http://dx.doi.org/10.15388/namc.2021.26.22356.

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We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single.

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GÁSPÁR, CSABA. "REGULARIZATION TECHNIQUES FOR THE METHOD OF FUNDAMENTAL SOLUTIONS." International Journal of Computational Methods 10, no. 02 (March 2013): 1341004. http://dx.doi.org/10.1142/s0219876213410041.

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A special regularization method based on higher-order partial differential equations is presented. Instead of using the fundamental solution of the original partial differential operator with source points located outside of the domain, the original second-order partial differential equation is approximated by a higher-order one, the fundamental solution of which is continuous at the origin. This allows the use of the method of fundamental solutions (MFS) for the approximate problem. Due to the continuity of the modified operator, the source points and the boundary collocation points are allowed to coincide, which makes the solution process simpler. This regularization technique is generalized to various problems and combined with the extremely efficient quadtree-based multigrid methods. Approximation theorems and numerical experiences are also presented.