A new clique polynomial approach for fractional partial differential equations

Author(s):
Kumbinarasaiah Srinivasa

Abstract This paper generates a novel approach called the clique polynomial method (CPM) using the clique polynomials raised in graph theory and used for solving the fractional order PDE. The fractional derivative is defined in terms of the Caputo fractional sense and the fractional partial differential equations (FPDE) are converted into nonlinear algebraic equations and collocated with suitable grid points in the current approach. The convergence analysis for the proposed scheme is constructed and the technique proved to be uniformly convegant. We applied the method for solving four problems to justify the proposed technique. Tables and graphs reveal that this new approach yield better results. Some theorems are discussed with proof.

2018 ◽
Vol 22 (Suppl. 1) ◽
pp. 277-286 ◽
Author(s):
Hossein Jafari ◽

In this work we suggest a numerical approach based on the B-spline polynomial to obtain the solution of linear fractional partial differential equations. We find the operational matrix for fractional integration and then we convert the main problem into a system of linear algebraic equations by using this matrix. Examples are provided to show the simplicity of our method.

2013 ◽
Vol 2013 ◽
pp. 1-13 ◽
Author(s):
Fukang Yin ◽
Junqiang Song ◽
Yongwen Wu ◽
Lilun Zhang

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

2019 ◽
Vol 2019 (1) ◽
Author(s):
S. Mockary ◽
E. Babolian ◽
A. R. Vahidi

Abstract In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

2013 ◽
Vol 2013 ◽
pp. 1-5 ◽
Author(s):
Fanwei Meng

We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1)-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained. This approach can be suitable for solving fractional partial differential equations with more general forms than the method proposed by S. Zhang and H.-Q. Zhang (2011).

2013 ◽
Vol 66 (5) ◽
pp. 838-843 ◽
Author(s):
H. Jafari ◽
M. Nazari ◽
D. Baleanu ◽
C.M. Khalique

2018 ◽
Vol 56 (6) ◽
pp. 2817-2828 ◽
Author(s):
Qinghua Feng

2018 ◽
Vol 15 (6) ◽
Author(s):
Soleiman Hosseinpour ◽
Alireza Nazemi ◽
Emran Tohidi

2014 ◽
Vol 2014 ◽
pp. 1-12 ◽
Author(s):
M. A. Mohamed ◽
M. Sh. Torky

The Legendre multiwavelet Galerkin method is adopted to give the approximate solution for the nonlinear fractional partial differential equations (NFPDEs). The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. We presented the numerical results and a comparison with the exact solution in the cases when we have an exact solution to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

2008 ◽
Vol 42 (6-8) ◽
pp. 639-651
Author(s):
Zaid Odibat ◽
Shaher Momani

2019 ◽
Vol 2019 (1) ◽
Author(s):
Di Xu ◽
Fanwei Meng

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.