Search

In this study, the Jacobi wavelet collocation method is studied to derive a solution of the time-fractional Fisher?s equation in Caputo sense. Jacobi wavelets can be considered as a generalization of the wavelets since the Gegenbauer, and thus also Chebyshev and Legendre polynomials are a special type of the Jacobi polynomials. So, more accurate and fast convergence solutions can be possible for some kind of problems thanks to Jacobi wavelets. After applying the proposed method to the considered equation and discretizing the equation at the collocation points, an algebraic equation system is derived and solving the equation system is quite sim?ple rather than solving a non-linear PDE. The obtained values of our method are checked against the other numerical and analytic solution of considered equation in the literature and the results are visualized by using graphics and tables so as to reveal whether the method is effectiveness or not. The obtained results evince that the wavelet method is quite proper because of its simple algorithm, high accuracy and less CPU time for solving the considered equation.

The main aim of this paper is to investigate the efficient Chebyshev wavelet collocation method for Ginzburg-Landau equation. The basic idea of this method is to have the approximation of Chebyshev wavelet series of a non-linear PDE. We demonstrate how to use the method for the numerical solution of the Ginzburg-Landau equation with initial and boundary conditions. For this purpose, we have obtained operational matrix for Chebyshev wavelets. By applying this technique in Ginzburg-Landau equation, the PDE is converted into an algebraic system of non-linear equations and this system has been solved using MAPLE computer algebra system. We demonstrate the validity and applicability of this technique which has been clarified by using an example. Exact solution is compared with an approximate solution. Moreover, Chebyshev wavelet collocation method is found to be acceptable, efficient, accurate and computational for the non-linear or PDE.

In this article, studied the properties of the oscillation of fractional difference equations, and we obtain some results. The results we obtained are an expansion and further development of highly known results. Then we showed them with examples.

This paper proposes a new technique which rests upon Legendre wavelets for solving linear and non-linear forms of fractional order initial and boundary value problems. In some particular circumstances, a new operational matrix of fractional derivative is generated by utilizing some significant properties of wavelets and orthogonal polynomials. We approached the solution in a finite series with respect to Legendre wavelets and then by using these operational matrices, we reduced the fractional differential equations into a system of algebraic equations. Finally, the introduced technique is tested on several illustrative examples. The obtained results demonstrate that this technique is a very impressive and applicable mathematical tool for solving fractional differential equations.

In this paper, the Legendre wavelet operational matrix method has been introduced for solving high-order linear and non-linear multi-point: initial and boundary value problems. It has been suggested that the technique is rest upon practical application of the operational matrix and its derivatives. The differential equation is presented that it is converted to a system of algebraic equations via the properties of Legendre wavelet together with the operational matrix method. As a result of this study, the scheme has been tested on five linear and non-linear problems. The results have demonstrated that this method is a very effective and advantageous tool in solving such problems.