The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations

Khalil, Hammad; Khan, Rahmat Ali

doi:10.1080/00207160.2014.945919

Cited by 22 publications

(18 citation statements)

References 17 publications

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“…Beside the fact that the AB type derivative has been recently introduced, many applications in this field can be found in the literature we talked about in the introduction section. One of many examples for modeling of natural phenomenon using AB type FDEs is the amount of drug lidocaine in the bloodstream and body tissue [47]. The human disease of ventricular arrythmia or irregular heartbeat is treated clinically using the drug lidocaine.…”

Section: Applicationmentioning

confidence: 99%

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

et al. 2018

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In this paper, we solve a system of fractional differential equations within a fractional derivative involving the Mittag-Leffler kernel by using the spectral methods. We apply the Chebyshev polynomials as a base and obtain the necessary operational matrix of fractional integral using the Clenshaw-Curtis formula. By applying the operational matrix, we obtain a system of linear algebraic equations. The approximate solution is computed by solving this system. The regularity of the solution investigated and a convergence analysis is provided. Numerical examples are provided to show the effectiveness and efficiency of the method.

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Section: Applicationmentioning

confidence: 99%

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

et al. 2018

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“…Example Consider the following SFDEs

D^{\frac{4}{5}} u_{1} false(x false) - u_{1} false(x false) + 3 u_{2} false(x false) = f_{1} false(x false),

D^{\frac{4}{5}} u_{2} false(x false) + 4 u_{1} false(x false) - 2 u_{2} false(x false) = f_{2} false(x false),

with initial conditions

u_{1} (0) = - 1, u_{2} (0) = - 1,

the source terms are defined as

f_{1} (x) = - x^{5} + \frac{15625 x^{\frac{21}{5}}}{924 Γ (\frac{1}{5})} + 4 x^{4} - \frac{625 x^{\frac{16}{5}}}{44 Γ (\frac{1}{5})} - 3 x^{3} - 2,

and

f_{2} (x) = 4 x^{5} - 6 x^{4} + \frac{625 x^{\frac{16}{5}}}{44 Γ (\frac{1}{5})} + 2 x^{3} - \frac{125 x^{\frac{11}{5}}}{11 Γ (\frac{1}{5})} - 2 .

The exact solution is u 1 ( x ) = x 5 − x 4 − 1 and u 2 ( x ) = x 4 − x 3 − 1.…”

Section: Test Problemsmentioning

confidence: 99%

“…Khalil and Khan used the shifted Jacobi operational matrix technique (SJOMT) for solving the above problem. Using the SJOMT, the error was about 10 −4 with N = 5 and was about 0 with N = 6, see the study of Khalil and Khan . We apply the technique described in Section at γ = δ = 0.5 with N = 4.…”

Section: Test Problemsmentioning

confidence: 99%

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Modified Galerkin algorithm for solving multitype fractional differential equations

Alsuyuti¹,

Doha

Ezz-Eldien

et al. 2019

Math Methods in App Sciences

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The primary point of this manuscript is to dissect and execute a new modified Galerkin algorithm based on the shifted Jacobi polynomials for solving fractional differential equations (FDEs) and system of FDEs (SFDEs) governed by homogeneous and nonhomogeneous initial and boundary conditions. In addition, we apply the new algorithm for solving fractional partial differential equations (FPDEs) with Robin boundary conditions and time‐fractional telegraph equation. The key thought for obtaining such algorithm depends on choosing trial functions satisfying the underlying initial and boundary conditions of such problems. Some illustrative examples are discussed to ascertain the validity and efficiency of the proposed algorithm. Also, some comparisons with some other existing spectral methods in the literature are made to highlight the superiority of the new algorithm.

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“…As much as we know they are not well studied and neither is properly applied to solve coupled systems of PFDEs. To the best of our knowledge very few articles are devoted to the numerical solutions of coupled systems of integral equations and ordinary and partial differential equations of fractional order; for details see [39][40][41].…”

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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The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations

Cited by 22 publications

References 17 publications

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel

Modified Galerkin algorithm for solving multitype fractional differential equations

Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

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