Mehmet Giyas Sakar

Variational iteration method for the time-fractional Fornberg-Whitham equation

This paper presents the approximate analytical solutions to solve the nonlinear Fornberg-Whitham equation with fractional time derivative. By using initial values, explicit solutions of the equations are solved by using a reliable algorithm like the variational iteration method. The fractional derivatives are taken in the Caputo sense. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of @a are presented graphically.

Iterative reproducing kernel Hilbert spaces method for Riccati differential equations

This paper presents iterative reproducing kernel Hilbert spaces method (IRKHSM) to obtain the numerical solutions for Riccati differential equations with constant and variable coefficients. Representation of the exact solution is given in the W 2 2 0 , X reproducing kernel space. Numerical solution of Riccati differential equations is acquired by interrupting the n -term of the exact solution. Also, the error of the numerical solution is monotone decreasing in terms of the norm of W 2 2 0 , X . The outcomes from numerical examples show that the present iterative algorithm is very effective and convenient.

Improving Variational Iteration Method with Auxiliary Parameter for Nonlinear Time-Fractional Partial Differential Equations

In this research, we present a new approach based on variational iteration method for solving nonlinear time-fractional partial differential equations in large domains. The convergence of the method is shown with the aid of Banach fixed point theorem. The maximum error bound is specified. The optimal value of auxiliary parameter is obtained by use of residual error function. The fractional derivatives are taken in the Caputo sense. Numerical examples that involve the time-fractional Burgers equation, the time-fractional fifth-order Korteweg–de Vries equation and the time-fractional Fornberg–Whitham equation are examined to show the appropriate properties of the method. The results reveal that a new approach is very effective and convenient.

Analytical Approximate Solutions of (n + 1)-Dimensional Fractal Heat-Like and Wave-Like Equations

In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems.