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Artigos de revistas sobre o tema "Numerical analysis of partial differential equation"

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Autor: Grafiati

Publicado: 13 de julho de 2024

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1

Alharthi, Nadiyah Hussain, Abdon Atangana e Badr S. Alkahtani. "Numerical analysis of some partial differential equations with fractal-fractional derivative". AIMS Mathematics 8, n.º 1 (2022): 2240–56. http://dx.doi.org/10.3934/math.2023116.

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<abstract> <p>In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed.</p> </abstract>

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2

Kurbonov, Elyorjon, Nodir Rakhimov, Shokhabbos Juraev e Feruza Islamova. "Derive the finite difference scheme for the numerical solution of the first-order diffusion equation IBVP using the Crank-Nicolson method". E3S Web of Conferences 402 (2023): 03029. http://dx.doi.org/10.1051/e3sconf/202340203029.

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In the article, a differential scheme is created for the the first-order diffusion equation using the Crank-Nicolson method. The stability of the differential scheme was checked using the Neumann method. To solve the problem numerically, stability intervals were found using the Neman method. This work presents an analysis of the stability of the Crank-Nicolson scheme for the two-dimensional diffusion equation using Von Neumann stability analysis. The Crank-Nicolson scheme is a widely used numerical method for solving partial differential equations that combines the explicit and implicit schemes. The stability analysis is an important factor to consider when choosing a numerical method for solving partial differential equations, as numerical instability can cause inaccurate solutions. We show that the Crank-Nicolson scheme is unconditionally stable, meaning that it can be used for a wide range of parameters without being affected by numerical instability. Overall, the analysis and implementation presented in this work provide a framework for designing and analyzing numerical methods for solving partial differential equations using the Crank-Nicolson scheme. The stability analysis is crucial for ensuring the accuracy and reliability of numerical solutions of partial differential equations.

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3

Sanz-Serna, J. M. "A Numerical Method for a Partial Integro-Differential Equation". SIAM Journal on Numerical Analysis 25, n.º 2 (abril de 1988): 319–27. http://dx.doi.org/10.1137/0725022.

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4

Zhao, J., M. S. Cheung e S. F. Ng. "Spline Kantorovich method and analysis of general slab bridge deck". Canadian Journal of Civil Engineering 25, n.º 5 (1 de outubro de 1998): 935–42. http://dx.doi.org/10.1139/l98-030.

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In this paper, the spline Kantorovich method is developed and applied to the analysis and design of bridge decks. First, the bridge deck is mapped into a unit square in the Xi - eta plane. The governing partial differential equation of the plate is reduced to the ordinary differential equation in the longitudinal direction of the bridge by the routine Kantorovich method. Spline point collocation method is then used to solve the derived ordinary differential equation to obtain the displacements and internal forces of the bridge deck. Mindlin plate theory is incorporated into the differential equation and, as a result, the effect of shear deformation of the plate is also considered. Possible shear locking is avoided by the reduced integration technique. Numerical examples show that the proposed new numerical model is versatile, efficient, and reliable.Key words: Kantorovich method, spline function, partial differential equations, ordinary differential equations, point collocation method, bridge deck.

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5

Pyanylo, Yaroslav, e Galyna Pyanylo. "Analysis of approaches to mass-transfer modeling n non-stationary mode". Physico-mathematical modelling and informational technologies, n.º 28, 29 (27 de dezembro de 2019): 55–64. http://dx.doi.org/10.15407/fmmit2020.28.055.

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A significant number of natural and physical processes are described by differential equations in partial derivatives or systems of differential equations in partial derivatives. Numerical methods have been found to find their solutions. Partial derivatives systems are solved mainly by reducing the order of the system of equations or reducing it to one differential equation. This procedure leads to an increase in the order of the differential equation. There are various restrictions and errors that can lead to additional solutions, boundary conditions for intermediate derivatives, and so on. The work is devoted to the analysis of such situations and ways of exit.

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6

Abrashina-Zhadaeva, N., e N. Romanova. "Vector Additive Decomposition for 2D Fractional Diffusion Equation". Nonlinear Analysis: Modelling and Control 13, n.º 2 (25 de abril de 2008): 137–43. http://dx.doi.org/10.15388/na.2008.13.2.14574.

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Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations. An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients. In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution.

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7

Reinfelds, Andrejs, Olgerts Dumbrajs, Harijs Kalis, Janis Cepitis e Dana Constantinescu. "NUMERICAL EXPERIMENTS WITH SINGLE MODE GYROTRON EQUATIONS". Mathematical Modelling and Analysis 17, n.º 2 (1 de abril de 2012): 251–70. http://dx.doi.org/10.3846/13926292.2012.662659.

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Gyrotrons are microwave sources whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. This process is described by the system of two complex differential equations: nonlinear first order ordinary differential equation with parameter (averaged equation of electron motion) and second order partial differential equation for high frequency field (RF field) in resonator (Schrödinger type equation for the wave amplitude). The stationary problem of the single mode gyrotron equation in short time interval with real initial conditions was numerically examined in our earlier work. In this paper we consider the stationary and nonstationary problems in large time interval with complex oscillating initial conditions. We use the implicit finite difference schemes and the method of lines realized with MATLAB. Two versions of gyrotron equation are investigated. We consider different methods for modelling new and old versions of the gyrotron equations. The main physical result is the possibility to determine the maximal value of the wave amplitude and the electron efficiency coefficient.

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8

Company, R., L. Jódar, M. Fakharany e M. C. Casabán. "Removing the Correlation Term in Option Pricing Heston Model: Numerical Analysis and Computing". Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/246724.

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This paper deals with the numerical solution of option pricing stochastic volatility model described by a time-dependent, two-dimensional convection-diffusion reaction equation. Firstly, the mixed spatial derivative of the partial differential equation (PDE) is removed by means of the classical technique for reduction of second-order linear partial differential equations to canonical form. An explicit difference scheme with positive coefficients and only five-point computational stencil is constructed. The boundary conditions are adapted to the boundaries of the rhomboid transformed numerical domain. Consistency of the scheme with the PDE is shown and stepsize discretization conditions in order to guarantee stability are established. Illustrative numerical examples are included.

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9

Kim, Sung-Hoon, e Youn-sik Park. "An Improved Finite Difference Type Numerical Method for Structural Dynamic Analysis". Shock and Vibration 1, n.º 6 (1994): 569–83. http://dx.doi.org/10.1155/1994/139352.

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An improved finite difference type numerical method to solve partial differential equations for one-dimensional (1-D) structure is proposed. This numerical scheme is a kind of a single-step, second-order accurate and implicit method. The stability, consistency, and convergence are examined analytically with a second-order hyperbolic partial differential equation. Since the proposed numerical scheme automatically satisfies the natural boundary conditions and at the same time, all the partial differential terms at boundary points are directly interpretable to their physical meanings, the proposed numerical scheme has merits in computing 1-D structural dynamic motion over the existing finite difference numeric methods. Using a numerical example, the suggested method was proven to be more accurate and effective than the well-known central difference method. The only limitation of this method is that it is applicable to only 1-D structure.

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10

Ratas, Mart, Andrus Salupere e Jüri Majak. "SOLVING NONLINEAR PDES USING THE HIGHER ORDER HAAR WAVELET METHOD ON NONUNIFORM AND ADAPTIVE GRIDS". Mathematical Modelling and Analysis 26, n.º 1 (18 de janeiro de 2021): 147–69. http://dx.doi.org/10.3846/mma.2021.12920.

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The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points.

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11

Turut, Veyis, e Nuran Güzel. "Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order". Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/746401.

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Two tecHniques were implemented, the Adomian decomposition method (ADM) and multivariate Padé approximation (MPA), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM), then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.

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12

Khattri, Sanjay Kumar. "Nonlinear elliptic problems with the method of finite volumes". Differential Equations and Nonlinear Mechanics 2006 (2006): 1–16. http://dx.doi.org/10.1155/denm/2006/31797.

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We present a finite volume discretization of the nonlinear elliptic problems. The discretization results in a nonlinear algebraic system of equations. A Newton-Krylov algorithm is also presented for solving the system of nonlinear algebraic equations. Numerically solving nonlinear partial differential equations consists of discretizing the nonlinear partial differential equation and then solving the formed nonlinear system of equations. We demonstrate the convergence of the discretization scheme and also the convergence of the Newton solver through a variety of practical numerical examples.

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13

Modanli, Mahmut, Bawar Mohammed Faraj e Faraedoon Waly Ahmed. "Using matrix stability for variable telegraph partial differential equation". An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 10, n.º 2 (1 de julho de 2020): 237–43. http://dx.doi.org/10.11121/ijocta.01.2020.00870.

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The variable telegraph partial differential equation depend on initial boundary value problem has been studied. The coefficient constant time-space telegraph partial differential equation is obtained from the variable telegraph partial differential equation throughout using Cauchy-Euler formula. The first and second order difference schemes were constructed for both of coefficient constant time-space and variable time-space telegraph partial differential equation. Matrix stability method is used to prove stability of difference schemes for the variable and coefficient telegraph partial differential equation. The variable telegraph partial differential equation and the constant coefficient time-space telegraph partial differential equation are compared with the exact solution. Finally, approximation solution has been found for both equations. The error analysis table presents the obtained numerical results.

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14

Wan-xie, Zhong, e Zhong Xiang-Xiang. "Elliptic partial differential equation and optimal control". Numerical Methods for Partial Differential Equations 8, n.º 2 (março de 1992): 149–69. http://dx.doi.org/10.1002/num.1690080206.

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15

Froese, Brittany D., e Adam M. Oberman. "Convergent Filtered Schemes for the Monge--Ampère Partial Differential Equation". SIAM Journal on Numerical Analysis 51, n.º 1 (janeiro de 2013): 423–44. http://dx.doi.org/10.1137/120875065.

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16

Woolway, Matthew, Byron A. Jacobs, Ebrahim Momoniat, Charis Harley e Dieter Britz. "Numerical Convergence Analysis of the Frank–Kamenetskii Equation". Entropy 22, n.º 1 (9 de janeiro de 2020): 84. http://dx.doi.org/10.3390/e22010084.

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This work investigates the convergence dynamics of a numerical scheme employed for the approximation and solution of the Frank–Kamenetskii partial differential equation. A framework for computing the critical Frank–Kamenetskii parameter to arbitrary accuracy is presented and used in the subsequent numerical simulations. The numerical method employed is a Crank–Nicolson type implicit scheme coupled with a fourth order spatial discretisation as well as a Newton–Raphson update step which allows for the nonlinear source term to be treated implicitly. This numerical implementation allows for the analysis of the convergence of the transient solution toward the steady-state solution. The choice of termination criteria, numerically dictating this convergence, is interrogated and it is found that the traditional choice for termination is insufficient in the case of the Frank–Kamenetskii partial differential equation which exhibits slow transience as the solution approaches the steady-state. Four measures of convergence are proposed, compared and discussed herein.

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17

Gavrilov, S. V., e A. M. Denisov. "Numerical Solution Methods for a Nonlinear Operator Equation Arising in an Inverse Coefficient Problem". Differential Equations 57, n.º 7 (julho de 2021): 868–75. http://dx.doi.org/10.1134/s0012266121070041.

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Abstract We consider the inverse problem of determining two unknown coefficients in a linear system of partial differential equations using additional information about one of the solution components. The problem is reduced to a nonlinear operator equation for one of the unknown coefficients. The successive approximation method and the Newton method are used to solve this operator equation numerically. Results of calculations illustrating the convergence of numerical methods for solving the inverse problem are presented.

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18

HIKIHARA, TAKASHI, KENTARO TORII e YOSHISUKE UEDA. "WAVE AND BASIN STRUCTURE IN SPATIALLY COUPLED MAGNETO-ELASTIC BEAM SYSTEM — TRANSITIONS BETWEEN COEXISTING WAVE SOLUTIONS". International Journal of Bifurcation and Chaos 11, n.º 04 (abril de 2001): 999–1018. http://dx.doi.org/10.1142/s0218127401002523.

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Standing and traveling waves are well-known phenomena of the coupled ordinary differential equations in many fields. The wave solutions of the coupled system are considered to be similar to the partial differential equation of the system. In this paper, the waves which appear in a coupled magneto-elastic beam system are discussed theoretically and numerically. The physical system is continuous elastically and discrete magnetically. There are several classes of models describing the system behavior. The Galerkin method is one of the powerful methods used to analyze the dynamics of the spatially distributed structure. The numerical solutions appearing in the coupled ordinary differential equation must show the spatially discrete characteristics even in the distributed system. However, most of the results obtained in the coupled systems are not more than the numerical approximation of the related partial differential equations. The large number of oscillators are given for the approximation. In this paper, the relationship between the coupled magneto-elastic beam system and the modified KdV equation is established by using the long wave approximation. However, in the short wavelength range, the approximation to the partial differential equation has no physical rationality. Therefore, the analysis of the difference–differential equation provides an important place of knowledge filling up the gap between the characteristics of the physical model and the numerical approximation.

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19

Emmrich, Etienne, e David Šiška. "Full discretisation of second-order nonlinear evolution equations: strong convergence and applications". Computational Methods in Applied Mathematics 11, n.º 4 (2011): 441–59. http://dx.doi.org/10.2478/cmam-2011-0025.

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Abstract Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

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20

Merdan, Mehmet, Ahmet Gökdoğan, Ahmet Yıldırım e 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.

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21

BENALI, ABDELKADER. "NUMERICAL RESOLUTION OF NON-LINEAR EQUATIONS". Journal of Science and Arts 23, n.º 3 (30 de setembro de 2023): 721–28. http://dx.doi.org/10.46939/j.sci.arts-23.3-a14.

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In this study, we have employed the highly significant hyperbolic tangent (tanh) method to conduct an in-depth analysis of nonlinear coupled KdV systems of partial differential equations. In comparison to existing sophisticated approaches, this proposed method yields more comprehensive exact solutions for traveling waves without requiring excessive additional effort. We have successfully applied this method to two examples drawn from the literature of nonlinear partial differential equation systems.

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22

ALdarawi, Iman, Banan Maayah, Eman Aldabbas e Eman Abuteen. "Numerical Solutions of Some Classes of Partial Differential Equations of Fractional Order". European Journal of Pure and Applied Mathematics 16, n.º 4 (30 de outubro de 2023): 2132–44. http://dx.doi.org/10.29020/nybg.ejpam.v16i4.4928.

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This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

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23

Yaxubayev, Kamil D., e Dinara D. Kochergina. "Numerical Analysis of the Exact Solution of the Wave Equation of the Longitudinal Seismic Oscillations of Soil and the Construction as a Point Insertion". Materials Science Forum 931 (setembro de 2018): 152–57. http://dx.doi.org/10.4028/www.scientific.net/msf.931.152.

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The numerical analysis of the exact solution of the system of the differential equations which includes the partial differential equation of the longitudinal seismic oscillations of the soil and the ordinary differential equation of oscillations of the construction in the form of a point rigid insertion.

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24

Cohn, Stephen E., e Dick P. Dee. "Observability of Discretized Partial Differential Equations". SIAM Journal on Numerical Analysis 25, n.º 3 (junho de 1988): 586–617. http://dx.doi.org/10.1137/0725037.

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25

Taghipour, M., e H. Aminikhah. "Pell Collocation Method for Solving the Nonlinear Time–Fractional Partial Integro–Differential Equation with a Weakly Singular Kernel". Journal of Function Spaces 2022 (23 de maio de 2022): 1–15. http://dx.doi.org/10.1155/2022/8063888.

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This article focuses on finding the numerical solution of the nonlinear time–fractional partial integro–differential equation. For this purpose, we use the operational matrices based on Pell polynomials to approximate fractional Caputo derivative, nonlinear, and integro–differential terms; and by collocation points, we transform the problem to a system of nonlinear equations. This nonlinear system can be solved by the fsolve command in Matlab. The method’s stability and convergence have been studied. Also included are five numerical examples to demonstrate the veracity of the suggested strategy.

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26

Bin Jebreen, Haifa, e Carlo Cattani. "Solving Time-Fractional Partial Differential Equation Using Chebyshev Cardinal Functions". Axioms 11, n.º 11 (14 de novembro de 2022): 642. http://dx.doi.org/10.3390/axioms11110642.

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We propose a numerical scheme based on the Galerkin method for solving the time-fractional partial differential equations. To this end, after introducing the Chebyshev cardinal functions (CCFs), using the relation between fractional integral and derivative, we represent the Caputo fractional derivative based on these bases and obtain an operational matrix. Applying the Galerkin method and using the operational matrix for the Caputo fractional derivative, the desired equation reduces to a system of linear algebraic equations. By solving this system, the unknown solution is obtained. The convergence analysis for this method is investigated, and some numerical simulations show the accuracy and ability of the technique.

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27

Murata, Souichi. "Nonclassical symmetry analysis for hyperbolic partial differential equation". Communications in Nonlinear Science and Numerical Simulation 13, n.º 8 (outubro de 2008): 1472–74. http://dx.doi.org/10.1016/j.cnsns.2006.10.009.

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28

Yan, Huahong. "Adaptive Wavelet Precise Integration Method for Nonlinear Black-Scholes Model Based on Variational Iteration Method". Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/735919.

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An adaptive wavelet precise integration method (WPIM) based on the variational iteration method (VIM) for Black-Scholes model is proposed. Black-Scholes model is a very useful tool on pricing options. First, an adaptive wavelet interpolation operator is constructed which can transform the nonlinear partial differential equations into a matrix ordinary differential equations. Next, VIM is developed to solve the nonlinear matrix differential equation, which is a new asymptotic analytical method for the nonlinear differential equations. Third, an adaptive precise integration method (PIM) for the system of ordinary differential equations is constructed, with which the almost exact numerical solution can be obtained. At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical result shows the method's higher numerical stability and precision.

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Shah, Kamal, Hafsa Naz, Muhammad Sarwar e Thabet Abdeljawad. "On spectral numerical method for variable-order partial differential equations". AIMS Mathematics 7, n.º 6 (2022): 10422–38. http://dx.doi.org/10.3934/math.2022581.

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<abstract><p>In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.</p></abstract>

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30

Tang, Quan, Ziyang Luo, Xindong Zhang e Juan Liu. "Analysis of Two-Level Mesh Method for Partial Integro-Differential Equation". Journal of Function Spaces 2022 (13 de julho de 2022): 1–10. http://dx.doi.org/10.1155/2022/4557844.

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In this paper, we present two-level mesh scheme to solve partial integro-differential equation. The proposed method is based on a finite difference method. For the first step, we use finite difference method in time and global radial basis function (RBF) finite difference (FD) in space. For the second step, we use the finite difference method to solve the proposed problem. This two-level mesh scheme is obtained by combining the radial basis function with finite difference. We prove the stability and convergence of scheme and show that the convergence order is O τ 2 + h 2 , where τ and h are the time step size and space step size, respectively. The results of numerical examples are compared with analytical solutions to show the efficiency of proposed scheme. The numerical results are in good agreement with theoretical ones.

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31

Hemeda, A. A. "Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method". Abstract and Applied Analysis 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/717540.

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An extension of the so-called new iterative method (NIM) has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

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Sultana, Mariam, Uroosa Arshad, Abdel-Haleem Abdel-Aty, Ali Akgül, Mona Mahmoud e Hichem Eleuch. "New Numerical Approach of Solving Highly Nonlinear Fractional Partial Differential Equations via Fractional Novel Analytical Method". Fractal and Fractional 6, n.º 9 (12 de setembro de 2022): 512. http://dx.doi.org/10.3390/fractalfract6090512.

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In this work, the fractional novel analytic method (FNAM) is successfully implemented on some well-known, strongly nonlinear fractional partial differential equations (NFPDEs), and the results show the approach’s efficiency. The main purpose is to show the method’s strength on FPDEs by minimizing the calculation effort. The novel numerical approach has shown to be the simplest technique for obtaining the numerical solution to any form of the fractional partial differential equation (FPDE).

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Chen, Yong, e Hongjun Gao. "Auxiliary equation method for solving nonlinear Wick-type partial differential equations". Communications in Nonlinear Science and Numerical Simulation 16, n.º 6 (junho de 2011): 2421–37. http://dx.doi.org/10.1016/j.cnsns.2010.09.014.

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Mourabit, A. El. "Parameters identification for a nonlinear partial differential equation in image denoising". Moroccan Journal of Pure and Applied Analysis 9, n.º 1 (1 de janeiro de 2023): 141–53. http://dx.doi.org/10.2478/mjpaa-2023-0010.

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Abstract In this work and in the context of PDE constrained optimization problems, we are interested in identification of a parameter in the diffusion equation proposed in [1]. We propose to identify this parameter automatically by a gradient descent algorithm to improve the restoration of a noisy image. Finally, we give numerical results to illustrate the performance of the automatic selection of this parameter and compare our numerical results with other image denoising approaches or algorithms.

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El Moutea, Omar, e Hassan El Amri. "Combined mixed finite element and nonconforming finite volume methods for flow and transport in porous media". Analysis 41, n.º 3 (23 de julho de 2021): 123–44. http://dx.doi.org/10.1515/anly-2018-0019.

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Abstract This paper is concerned with numerical methods for a coupled system of two partial differential equations (PDEs), modeling flow and transport of a contaminant in porous media. This coupled system, arising in modeling of flow and transport in heterogeneous porous media, includes two types of equations: an elliptic and a diffusion-convection equation. We focus on miscible flow in heterogeneous porous media. We use the mixed finite element method for the Darcy flow equation over triangles, and for the concentration equation, we use nonconforming finite volume methods in unstructured mesh. Finally, we show the existence and uniqueness of a solution of this coupled scheme and demonstrate the effectiveness of the methodology by a series of numerical examples.

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Yserentant, Harry. "Old and new convergence proofs for multigrid methods". Acta Numerica 2 (janeiro de 1993): 285–326. http://dx.doi.org/10.1017/s0962492900002385.

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Multigrid methods are the fastest known methods for the solution of the large systems of equations arising from the discretization of partial differential equations. For self-adjoint and coercive linear elliptic boundary value problems (with Laplace's equation and the equations of linear elasticity as two typical examples), the convergence theory reached a mature, if not its final state. The present article reviews old and new developments for this type of equation and describes the recent advances.

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Lord, Gabriel J., e Tony Shardlow. "Postprocessing for Stochastic Parabolic Partial Differential Equations". SIAM Journal on Numerical Analysis 45, n.º 2 (janeiro de 2007): 870–89. http://dx.doi.org/10.1137/050640138.

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Aghili, Arman. "Non-homogeneous impulsive time fractional heat conduction equation". Journal of Numerical Analysis and Approximation Theory 52, n.º 1 (10 de julho de 2023): 22–33. http://dx.doi.org/10.33993/jnaat521-1316.

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This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.

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Zheng, Jiachun, e Yunlei Yang. "M-WDRNNs: Mixed-Weighted Deep Residual Neural Networks for Forward and Inverse PDE Problems". Axioms 12, n.º 8 (30 de julho de 2023): 750. http://dx.doi.org/10.3390/axioms12080750.

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Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations in recent years. But studies have shown that there is a gradient pathology in PINNs. That is, there is an imbalance gradient problem in each regularization term during back-propagation, which makes it difficult for neural network models to accurately approximate partial differential equations. Based on the depth-weighted residual neural network and neural attention mechanism, we propose a new mixed-weighted residual block in which the weighted coefficients are chosen autonomously by the optimization algorithm, and one of the transformer networks is replaced by a skip connection. Finally, we test our algorithms with some partial differential equations, such as the non-homogeneous Klein–Gordon equation, the (1+1) advection–diffusion equation, and the Helmholtz equation. Experimental results show that the proposed algorithm significantly improves the numerical accuracy.

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Munoz, Ana Isabel, e Jose Ignacio Tello. "MATHEMATICAL ANALYSIS AND NUMERICAL SIMULATION IN MAGNETIC RECORDING". Mathematical Modelling and Analysis 19, n.º 3 (1 de junho de 2014): 334–46. http://dx.doi.org/10.3846/13926292.2014.924081.

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The head-tape interaction in magnetic recording is described in the literature by a coupled system of partial differential equations. In this paper we study the limit case of the system which reduces the problem to a second order nonlocal equation on a one-dimensional domain. We describe the numerical method of resolution of the problem, which is reformulated as an obstacle one to prevent head-tape contact. A finite element method and a duality algorithm handling Yosida approximation tools for maximal monotone operators are used in order to solve numerically the obstacle problem. Numerical simulations are introduced to describe some qualitative properties of the solution. Finally some conclusions are drawn.

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Zhu, Yuanye. "Quantum-Solving Algorithm for d’Alembert Solutions of the Wave Equation". Entropy 25, n.º 1 (29 de dezembro de 2022): 62. http://dx.doi.org/10.3390/e25010062.

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When faced with a quantum-solving problem for partial differential equations, people usually transform such problems into Hamiltonian simulation problems or quantum-solving problems for linear equation systems. In this paper, we propose a third approach to solving partial differential equations that differs from the two approaches. By using the duality quantum algorithm, we construct a quantum-solving algorithm for solving the first-order wave equation, which represents a typical class of partial differential equations. Numerical results of the quantum circuit have high precision consistency with the theoretical d’Alembert solution. Then the routine is applied to the wave equation with either a dissipation or dispersion term. As shown by complexity analysis for all these cases of the wave equation, our algorithm has a quadratic acceleration for each iteration compared to the classical algorithm.

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Cebers, Andrejs, e Harijs Kalis. "NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD". Mathematical Modelling and Analysis 18, n.º 1 (1 de fevereiro de 2013): 80–96. http://dx.doi.org/10.3846/13926292.2013.756835.

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Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions is obtained using method of lines for solving a large system of ordinary differential equations (ODE).

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Bachmayr, Markus. "Low-rank tensor methods for partial differential equations". Acta Numerica 32 (maio de 2023): 1–121. http://dx.doi.org/10.1017/s0962492922000125.

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Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

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Otto, Kurt. "Analysis of Preconditioners for Hyperbolic Partial Differential Equations". SIAM Journal on Numerical Analysis 33, n.º 6 (dezembro de 1996): 2131–65. http://dx.doi.org/10.1137/s0036142994264894.

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Yang, Miaomiao, Wentao Ma e Yongbin Ge. "BARYCENTRIC RATIONAL INTERPOLATION METHOD OF THE HELMHOLTZ EQUATION WITH IRREGULAR DOMAIN". Mathematical Modelling and Analysis 28, n.º 2 (21 de março de 2023): 330–51. http://dx.doi.org/10.3846/mma.2023.16408.

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In the work, a numerical method of the 2D Helmholtz equation with meshless interpolation collocation method is developed, which is defined in arbitrary domain with irregular shape. In our numerical method, based on the Chebyshev points, the partial derivatives and the spatial variables are discretized by the barycentric rational form basis function. After that the differential equations are simplified by employing differential matrix. To verify the the accuracy, effectiveness and stability in our method, some numerical tests based on the three types of different test points are adopted. Moreover, we can also verify that present method can be applied to both variable wave number problems and high wave number problems.

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Wu, Fangfang, Chuangui Lu, Yingying Wang e Na Hu. "Lattice Boltzmann Model for a Class of Time Fractional Partial Differential Equation". Axioms 12, n.º 10 (11 de outubro de 2023): 959. http://dx.doi.org/10.3390/axioms12100959.

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This paper is concerned with the lattice Boltzmann (LB) method for a class of time fractional partial differential equations (FPDEs) in the Caputo sense. By utilizing the properties of the Caputo derivative and discretization in time, FPDEs can be approximately transformed into standard partial differential equations with integer orders. Through incorporating an auxiliary distribution function into the evolution equation, which assists in recovering the macroscopic quantity u, the LB model with spatial second-order accuracy is constructed. The numerical experiments verify that the numerical results are in good agreement with analytical solutions and that the accuracy of the present model is better than the previous solutions.

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Yuan, Yirang, Qing Yang, Changfeng Li e Tongjun Sun. "A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis". Numerical Mathematics: Theory, Methods and Applications 10, n.º 3 (20 de junho de 2017): 541–61. http://dx.doi.org/10.4208/nmtma.2017.y15013.

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AbstractA coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

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Blanco, Pablo, Paola Gervasio e Alfio Quarteroni. "Extended Variational Formulation for Heterogeneous Partial Differential Equations". Computational Methods in Applied Mathematics 11, n.º 2 (2011): 141–72. http://dx.doi.org/10.2478/cmam-2011-0008.

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AbstractWe address the coupling of an advection equation with a diffusion-advection equation, for solutions featuring boundary layers. We consider non-overlapping domain decompositions and we face up the heterogeneous problem using an extended variational formulation. We will prove the equivalence between the latter formulation and a treatment based on a singular perturbation theory. An exhaustive comparison in terms of solution and computational efficiency between these formulations is carried out.

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Leng, Yu, Lampros Svolos, Dibyendu Adak, Ismael Boureima, Gianmarco Manzini, Hashem Mourad e Jeeyeon Plohr. "A guide to the design of the virtual element methods for second- and fourth-order partial differential equations". Mathematics in Engineering 5, n.º 6 (2023): 1–22. http://dx.doi.org/10.3934/mine.2023100.

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<abstract><p>We discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by <sup>[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>,<xref ref-type="bibr" rid="b3">3</xref>]</sup>, we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use $ C^0 $ and $ C^1 $ virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.</p></abstract>

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Bellen, Alfredo, Stefano Maset, Marino Zennaro e Nicola Guglielmi. "Recent trends in the numerical solution of retarded functional differential equations". Acta Numerica 18 (maio de 2009): 1–110. http://dx.doi.org/10.1017/s0962492906390010.

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Retarded functional differential equations (RFDEs) form a wide class of evolution equations which share the property that, at any point, the rate of the solution depends on a discrete or distributed set of values attained by the solution itself in the past. Thus the initial problem for RFDEs is an infinite-dimensional problem, taking its theoretical and numerical analysis beyond the classical schemes developed for differential equations with no functional elements. In particular, numerically solving initial problems for RFDEs is a diffcult task that cannot be founded on the mere adaptation of well-known methods for ordinary, partial or integro-differential equations to the presence of retarded arguments. Indeed, efficient codes for their numerical integration need speciffc approaches designed according to the nature of the equation and the behaviour of the solution.By defining the numerical method as a suitable approximation of the solution map of the given equation, we present an original and unifying theory for the convergence and accuracy analysis of the approximate solution. Two particular approaches, both inspired by Runge–Kutta methods, are described. Despite being apparently similar, they are intrinsically different. Indeed, in the presence of speciffc types of functionals on the right-hand side, only one of them can have an explicit character, whereas the other gives rise to an overall procedure which is implicit in any case, even for non-stiff problems.In the panorama of numerical RFDEs, some critical situations have been recently investigated in connection to speciffc classes of equations, such as the accurate location of discontinuity points, the termination and bifurcation of the solutions of neutral equations, with state-dependent delays, the regularization of the equation and the generalization of the solution behind possible termination points, and the treatment of equations stated in the implicit form, which include singularly perturbed problems and delay differential-algebraic equations as well. All these issues are tackled in the last three sections.In this paper we have not considered the important issue of stability, for which we refer the interested reader to the comprehensive book by Bellen and Zennaro (2003).

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