A Linear B-Spline Approximation for a Class of Nonlinear Time and Space Fractional Partial Differential Equations

2020 ◽  
pp. 67-85
Author(s):  
Behrouz Parsa Moghaddam ◽  
José António Tenreiro Machado ◽  
Arman Dabiri
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Spline Approximation ◽  
Fractional Partial Differential Equations ◽  
Time And Space ◽  
Partial Differential ◽  
B Spline ◽  
Linear B

B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations

2017 ◽  
Vol 15 (04) ◽  
pp. 1750034 ◽  
Author(s):  
Zeynab Kargar ◽  
Habibollah Saeedi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Results ◽  
Scaling Functions ◽  
Operational Matrices ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
B Spline ◽  
Time Space ◽  
Linear B

In this paper, the linear B-spline scaling functions and wavelets operational matrix of fractional integration are derived. A new approach implementing the linear B-spline scaling functions and wavelets operational matrices combining with the spectral tau method is introduced for approximating the numerical solutions of time-space fractional partial differential equations with initial-boundary conditions. They are utilized to reduce the main problem to a system of algebraic equations. The uniform convergence analysis for the linear B-spline scaling functions and wavelets expansion and an efficient error estimation of the presented method are also introduced. Illustrative examples are given and numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Special attention is given to a comparison between the numerical results obtained by our new technique and those found by other known methods.

scholarly journals Invariant subspace method: A tool for solving fractional partial differential equations

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Sangita Choudhary ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Caputo Derivative ◽  
Linear Time ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Time And Space ◽  
Partial Differential ◽  
Non Linear

AbstractIn this paper invariant subspace method has been employed for solving linear and non-linear time and space fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the effectiveness and applicability of the method.

scholarly journals Solving systems of multi-term fractional PDEs: Invariant subspace approach

2019 ◽  
Vol 10 (01) ◽  
pp. 1941010 ◽  
Author(s):  
Sangita Choudhary ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Time And Space ◽  
Partial Differential ◽  
Fractional Pdes

In the present paper, invariant subspace method has been extended for solving systems of multi-term fractional partial differential equations (FPDEs) involving both time and space fractional derivatives. Further, the method has also been employed for solving multi-term fractional PDEs in [Formula: see text] dimensions. A diverse set of examples is solved to illustrate the method.

A modified method for solving non-linear time and space fractional partial differential equations

2019 ◽  
Vol 36 (7) ◽  
pp. 2162-2178
Author(s):  
Umer Saeed ◽  
Muhammad Umair
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Time ◽  
Differential Quadrature Method ◽  
Fractional Partial Differential Equations ◽  
Lagrange Polynomial ◽  
Time And Space ◽  
Partial Differential ◽  
Content Type ◽  
Non Linear

Purpose The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain. Design/methodology/approach The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations. Findings The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed. Originality/value Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

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.

Sign in / Sign up

Export Citation Format

Share Document