Smoothing of Crank–Nicolson scheme for the two-dimensional diffusion with an integral condition

Siddique, Mohammad

doi:10.1016/j.amc.2009.04.025

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…Such formulation is placed in a separate class known as nonlocal boundary value problems. Some of the numerical investigations regarding PDEs with nonlocal boundary conditions reported in the literature can be found in [8][9][10][11][12][13][14][15]. Among others, some of the well-known methods that can be effectively applied to BVPs are finite difference methods, mesh-free methods, finite element methods, etc.…”

Section: Introductionmentioning

confidence: 99%

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A Novel Method for Solution of Fractional Order Two-Dimensional Nonlocal Heat Conduction Phenomena

Khalil

Hashim

Khan

et al. 2021

Mathematical Problems in Engineering

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In this paper, we have extended the operational matrix method for approximating the solution of the fractional-order two-dimensional elliptic partial differential equations (FPDEs) under nonlocal boundary conditions. We use a general Legendre polynomials basis and construct some new operational matrices of fractional order operations. These matrices are used to convert a sample nonlocal heat conduction phenomenon of fractional order to a structure of easily solvable algebraic equations. The solution of the algebraic structure is then used to approximate a solution of the heat conduction phenomena. The proposed method is applied to some test problems. The obtained results are compared with the available data in the literature and are found in good agreement.Dedicated to my father Mr. Sher Mumtaz, (1955-2021), who gave me the basic knowledege of mathematics.

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Section: Introductionmentioning

confidence: 99%

“…For instance, the one-dimensional heat equation with nonlocal boundary conditions has been studied in [12,[16][17][18]. Two-dimensional diffusion problems with nonlocal boundary conditions have been discussed in [13,19]. e numerical solution of the Laplace equation with integral boundary condition is explored in [8].…”

Section: Introductionmentioning

confidence: 99%

A Novel Method for Solution of Fractional Order Two-Dimensional Nonlocal Heat Conduction Phenomena

Khalil

Hashim

Khan

et al. 2021

Mathematical Problems in Engineering

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“…Nonlocal FDEs arise in mathematical modeling of various problems in physics, engineering, ecology, and biological sciences [ 28 , 29 , 30 ]. Some of the numerical investigations regarding FDEs with nonlocal constrains are discussed in [ 31 , 32 , 33 , 34 , 35 ]. Numerical approaches such as finite difference and radial base function also remain a focus of interest.…”

Section: Introductionmentioning

confidence: 99%

“…Application of these methods to one-dimensional heat-like equations has been studied in [ 32 , 36 , 37 , 38 ]. Two-dimensional diffusion problems [ 33 , 39 , 40 ] and Laplace equations with integral constraints are explored in [ 31 ].…”

Section: Introductionmentioning

confidence: 99%

Extension of Operational Matrix Technique for the Solution of Nonlinear System of Caputo Fractional Differential Equations Subjected to Integral Type Boundary Constrains

Khalil

Hashim

et al. 2021

Entropy

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We extend the operational matrices technique to design a spectral solution of nonlinear fractional differential equations (FDEs). The derivative is considered in the Caputo sense. The coupled system of two FDEs is considered, subjected to more generalized integral type conditions. The basis of our approach is the most simple orthogonal polynomials. Several new matrices are derived that have strong applications in the development of computational scheme. The scheme presented in this article is able to convert nonlinear coupled system of FDEs to an equivalent S-lvester type algebraic equation. The solution of the algebraic structure is constructed by converting the system into a complex Schur form. After conversion, the solution of the resultant triangular system is obtained and transformed back to construct the solution of algebraic structure. The solution of the matrix equation is used to construct the solution of the related nonlinear system of FDEs. The convergence of the proposed method is investigated analytically and verified experimentally through a wide variety of test problems.

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“…M. Siddique presented Pade schemes [31] and a third order L 0 −stable numerical scheme [27] for the numerical solution of problem (1)- (7). Authors of [24,25,26], proposed some numerical solution to the (1)- (7). The method is based on finding a solution in the form of a polynomial in three…”

Section: Introductionmentioning

confidence: 99%

Solution of the Two-Dimensional Second-Order Diffusion Equation With Nonlocal Boundary Condition

Afshar¹,

Soltanalizadeh²

2014

Int. J. of Pure and Appl. Math.

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Smoothing of Crank–Nicolson scheme for the two-dimensional diffusion with an integral condition

Cited by 6 publications

References 16 publications

A Novel Method for Solution of Fractional Order Two-Dimensional Nonlocal Heat Conduction Phenomena

A Novel Method for Solution of Fractional Order Two-Dimensional Nonlocal Heat Conduction Phenomena

Extension of Operational Matrix Technique for the Solution of Nonlinear System of Caputo Fractional Differential Equations Subjected to Integral Type Boundary Constrains

Solution of the Two-Dimensional Second-Order Diffusion Equation With Nonlocal Boundary Condition

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