A New Method to Solve Dual Systems of Fractional Integro-Differential Equations by Legendre Wavelets

The method that will be presented, is numerical solution based on the Legendre wavelets for solving dual systems of fractional integro-differential equations (FIDEs). First of all we make the operational matrix of fractional order integration. The application of this matrix is transforming FIDEs to a system of algebric equations. By this changing, we are able to solve it by a simple solution. In this way, the Legendre wavelets and their operator matrix are the most important keys of our solution. After explaining the method we test on some illustrative examples which numerical solutions of these examples demonstrate the validity and applicability of suggested method.

Convergence Analysis of Legendre wavelets in numerical solution of linear weakly singular Volterra integral equation for union of some intervals with application in heat conduction

In this paper we apply the Legendre wavelets basis to solve the linear weakly singular Volterra integral equation of the second kind. The basis is defined on [0,1) , and in this work we extend this interval to [0,n) for some positive integer n. For this aim we solve the problem on [0,1); then we apply the Legendre wavelets on [1,2) and use the lag solution on [0,1) to obtain the solution on [0,2) and continue this procedure. Convergence analysis of Legendre wavelets on [n,n+1), is considered in Section2. We give a convergence analysis for the proposed method, established on compactness of operators. In numerical results we give two sample problems from heat conduction. For this purpose, in Section 6 we give an equivalent theorem between the proposed heat conduction problem and an integral equation. Then we solve the equivalent integral equation by the proposed method on union of some interval and obtain the solution of the heat conduction problem. As Tables and Figures of two and three dimensional plots show, accuracy of the method is reasonable and there is not any propagation of error from lag intervals. The convergence analysis and these sample problems demonstrate the accuracy and applicability of the method.

A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations

In the present paper, fractional-order generalized Legendre wavelets (FOGLWs) are introduced. We apply the FOGLWs for solving fractional Riccati differential equation. By using the hypergeometric function, we obtain an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FOGLWs. By using this exact formula and the properties of the FOGLWs, we reduce the solution of the fractional Riccati differential equation to the solution of an algebraic system. This algebraic system can be solved effectively. This method gives very accurate results. The given numerical examples support this claim.