Convergence of numerical schemes for the solution of partial integro-differential equations used in heat transfer

Al‐Khaled, Kamel; Darweesh, Amer; Yousef, Maha

doi:10.1007/s12190-019-01268-9

Cited by 10 publications

(6 citation statements)

References 17 publications

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“…Applying the traveling wave transform (14) to the above equation and integrating the resulting equation once, we then obtain the following ODE in V(ξ) as…”

Section: Exact Solutions For the (2 + 1)-dimensional Conformable Time Partial Integro-differential Jm Evolution Equation Using The Generamentioning

confidence: 99%

“…The initial and boundary value problem expressed by the nonlinear weakly singular partial integro-differential equation arising from viscoelasticity was proposed and analyzed using a Legendre wavelet collocation method (LWCM) by Singh et al [13]. Khaled [14] employed the sinc-Galerkin method to obtain numerical solutions for a parabolic Volterra integro-differential equation presenting the heat transfer of heterogeneous materials. To further grasp other applications of PIDEs, one can refer to [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the investigation of finding solutions to those PIDEs plays an important role in describing mechanisms and physical behaviors of the unknown variables in such problems. Some recent examples of the methods that have been employed to obtain numerical solutions and approximate analytical solutions for PIDEs can be found in [13,14,20,21]. However, the behaviors of the systems formulated using PIDEs can be precisely described via their exact solutions.…”

Section: Introductionmentioning

confidence: 99%

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Application of the Exp-Function and Generalized Kudryashov Methods for Obtaining New Exact Solutions of Certain Nonlinear Conformable Time Partial Integro-Differential Equations

2021

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The key objective of this paper is to construct exact traveling wave solutions of the conformable time second integro-differential Kadomtsev–Petviashvili (KP) hierarchy equation using the Exp-function method and the (2 + 1)-dimensional conformable time partial integro-differential Jaulent–Miodek (JM) evolution equation utilizing the generalized Kudryashov method. These two problems involve the conformable partial derivative with respect to time. Initially, the conformable time partial integro-differential equations can be converted into nonlinear ordinary differential equations via a fractional complex transformation. The resulting equations are then analytically solved via the corresponding methods. As a result, the explicit exact solutions for these two equations can be expressed in terms of exponential functions. Setting some specific parameter values and varying values of the fractional order in the equations, their 3D, 2D, and contour solutions are graphically shown and physically characterized as, for instance, a bell-shaped solitary wave solution, a kink-type solution, and a singular multiple-soliton solution. To the best of the authors’ knowledge, the results of the equations obtained using the proposed methods are novel and reported here for the first time. The methods are simple, very powerful, and reliable for solving other nonlinear conformable time partial integro-differential equations arising in many applications.

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“…Applying the traveling wave transform (14) to the above equation and integrating the resulting equation once, we then obtain the following ODE in V(ξ) as…”

Section: Exact Solutions For the (2 + 1)-dimensional Conformable Time Partial Integro-differential Jm Evolution Equation Using The Generamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Application of the Exp-Function and Generalized Kudryashov Methods for Obtaining New Exact Solutions of Certain Nonlinear Conformable Time Partial Integro-Differential Equations

2021

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“…Over the past three decades, various numerical methods based on the Sinc approximation have been presented, which have the advantages of a very fast convergence of exponential order and handling singularities effectively. The Sinc method proposed by Frank Stenger [35][36][37] has been increasingly applied to solve a variety of linear and nonlinear models that arise in scientific and engineering applications such as two-point boundary value problems [38], the Blasius equation [39], oceanographic problems with boundary layers [40], fourth-order partial integro-differential equation [41], the Volterra integro-differential equation [42,43], optimal control, heat distribution and astrophysics equations. According to the definition, it can be seen that fractional derivatives and integrals always deal with weak singularities.…”

Section: Introductionmentioning

confidence: 99%

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

Li,

Chen,

Zhou

2023

Axioms

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In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method is applied for discretizing the spatial derivative.The exponential convergence of our proposed method is demonstrated in detail. Finally, numerical evidence is employed to verify the theoretical results and confirm the expected convergence rate.

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“…The study of nonlinear partial integro-differential equations (NPIDEs) has become a subject of considerable interest. These equations and similar ones arise in a variety of science and technology fields such as reaction-diffusion problems ( [13]), theory of elasticity ( [23]), heat conduction ( [1]), mechanic of solids ( [2]), population dynamics ( [20]), transient, conductive and radiative transport ( [14]), and other applications.…”

Section: Introductionmentioning

confidence: 99%

Projected Iterations of Fixed-Point Type to Solve Nonlinear Partial Volterra Integro-Differential Equations

Berenguer

Gámez

2020

Bull. Malays. Math. Sci. Soc.

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In this paper, we propose a method to approximate the fixed point of an operator in a Banach space.Using biorthogonal systems, this method is applied to build an approximation of the solution of a class of nonlinear partial integro-differential equations. The theoretical findings are illustrated with several numerical examples, confirming the reliability, validity and precision of the proposed method.

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Convergence of numerical schemes for the solution of partial integro-differential equations used in heat transfer

Cited by 10 publications

References 17 publications

Application of the Exp-Function and Generalized Kudryashov Methods for Obtaining New Exact Solutions of Certain Nonlinear Conformable Time Partial Integro-Differential Equations

Application of the Exp-Function and Generalized Kudryashov Methods for Obtaining New Exact Solutions of Certain Nonlinear Conformable Time Partial Integro-Differential Equations

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

Projected Iterations of Fixed-Point Type to Solve Nonlinear Partial Volterra Integro-Differential Equations

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