A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation

Khalil, Hammad; Khan, Rahmat Ali

doi:10.1016/j.camwa.2014.03.008

Cited by 75 publications

(43 citation statements)

References 13 publications

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“…Since the shifted Legendre operational matrix for the variable-order differentiation is upper triangular matrix, compared with the two-dimensional Legendre operational matrix for the fractional differentiation in [52], it greatly reduces the memory space.…”

Section: Theorem 31 the Caputo Variable-order Fractional Derivative mentioning

confidence: 99%

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

206

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The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one-and twodimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.Keywords One-dimensional cable equation · Two-dimensional variable-order nonlinear cable equation · Collocation method · Jacobi polynomials · Operational matrix of fractional derivative · Variable-order derivative

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Section: Theorem 31 the Caputo Variable-order Fractional Derivative mentioning

confidence: 99%

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Bhrawy

Zaky²

2014

Nonlinear Dyn

206

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“…In [37], the authors established a new numerical scheme based on the Haar wavelet for solving (FPDEs). More recently, the authors [38] developed some new results related to the Jacobi polynomials for operational matrices. All the operational matrices methods are used to solve FDEs and FPDEs.…”

Section: Introductionmentioning

confidence: 99%

“…It is remarkable that the proposed system contains the coupled system investigated in [38] as a special case. If we take ( , ) = ( , ) = 0 together with 1 = 0, 2 = 1 and 1 = 0, 2 = 1, the proposed system becomes a coupled system of Laplace equations of fractional order by taking…”

Section: Introductionmentioning

confidence: 99%

Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

Shah

2017

Advances in Mathematical Physics

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We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

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“…In the case of a single variable, the operational matrices of fractional-order integration and differentiation are available in [42,49]. Similarly, operational matrices for two-dimensional orthogonal polynomials can be found in [22][23][24]. We generalize the notion to the case of two variables and develop operational matrices of fractional-order integrations and differentiations.…”

Section: Operational Matrices Of Integrations and Differentiationsmentioning

confidence: 99%

“…In continuation, a new numerical scheme, based on the Haar wavelet, involving an operational matrix of integration is developed in [4] for solutions of fractional-order multi-point boundary value problems. Some new results related to the Jacobi polynomials and operational matrices have been recently discovered (see [22][23][24]). Some operational matrices of arbitrary-order derivatives and integrals and their applications are also constructed by using B-spline functions, fractional Jacobi functions and the Taylor series method (for details, see [9,21,27]).…”

Section: Introductionmentioning

confidence: 99%

A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations

Shah¹,

Khalil²,

Khan³

2017

LMS J. Comput. Math.

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Due to the increasing application of fractional calculus in engineering and biomedical processes, we analyze a new method for the numerical simulation of a large class of coupled systems of fractional-order partial differential equations. In this paper, we study shifted Jacobi polynomials in the case of two variables and develop some new operational matrices of fractional-order integrations as well as fractional-order differentiations. By the use of these operational matrices, we present a new and easy method for solving a generalized class of coupled systems of fractionalorder partial differential equations subject to some initial conditions. We convert the system under consideration to a system of easily solvable algebraic equation without discretizing the system, and obtain a highly accurate solution. Also, the proposed method is compared with some other well-known differential transform methods. The proposed method is computer oriented. We use MatLab to perform the necessary calculation. The next two parts will appear soon.

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A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation

Cited by 75 publications

References 13 publications

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations

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