scholarly journals Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

2021 ◽  
Vol 5 (4) ◽  
pp. 208
Author(s):  
Muhammad I. Bhatti ◽  
Md. Habibur Rahman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Order ◽  
Initial Conditions ◽  
Operational Matrix ◽  
Basis Sets ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

scholarly journals Approximate Solution of Fractional Nonlinear Partial Differential Equations by the Legendre Multiwavelet Galerkin Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
M. A. Mohamed ◽  
M. Sh. Torky
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Partial Differential Equations ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

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.

scholarly journals Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Weishi Yin ◽  
Fei Xu ◽  
Weipeng Zhang ◽  
Yixian Gao
Keyword(s):  
Asymptotic Expansion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Power Series Method ◽  
Time Space ◽  
Residual Power Series ◽  
Asymptotic Expansion Of Solutions

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

scholarly journals Numerical Methods for Fractional Order Singular Partial Differential Equations with Variable Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Variational Iteration Method ◽  
Convergent Series ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

We implement relatively analytical methods, the homotopy perturbation method and the variational iteration method, for solving singular fractional partial differential equations of fractional order. The process of the methods which produce solutions in terms of convergent series is explained. The fractional derivatives are described in Caputo sense. Some examples are given to show the accurate and easily implemented of these methods even with the presence of singularities.

scholarly journals Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases

2021 ◽  
Vol 5 (3) ◽  
pp. 106
Author(s):  
Muhammad Bhatti ◽  
Md. Rahman ◽  
Nicholas Dimakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Matrix Equation ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Initial Guess ◽  
Partial Differential

A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients ai(t) and B-polys Bi(x) has been employed. Additionally, the variable quantities in the anticipated solution are determined using the Galerkin method for minimizing errors. Before the minimization process is to take place, the NPDE is converted into an operational matrix equation which, when inverted, yields values of the undefined coefficients in the expected solution. The nonlinear terms of the NPDE are combined in the operational matrix equation using the initial guess and iterated until converged values of coefficients are obtained. A valid converged solution of NPDE is established when an appropriate degree of B-poly basis is employed, and the initial conditions are imposed on the operational matrix before the inverse is invoked. However, the accuracy of the solution depends on the number of B-polys of a certain degree expressed in multidimensional variables. Four examples of NPDE have been worked out to show the efficacy and accuracy of the two-dimensional B-poly technique. The estimated solutions of the examples are compared with the known exact solutions and an excellent agreement is found between them. In calculating the solutions of the NPD equations, the currently employed technique provides a higher-order precision compared to the finite difference method. The present technique could be readily extended to solving complex partial differential equations in multivariable problems.

scholarly journals New method for solving a class of fractional partial differential equations with applications

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 277-286 ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations ◽  
Spline Polynomial

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.

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