SOLUTIONS FOR THE FRACTIONAL MATHEMATICAL MODELS OF DIFFUSION PROCESS

In this research we present two new approaches with Laplace transformation to form the truncated solution of space-time fractional differential equations (STFDE) with mixed boundary conditions. Since order of the fractional derivative of time derivative is taken between zero and one we have a sub-diffusive differential equation. First, we reduce STFDE into either a time or a space fractional differential equation which are easier to deal with. At the second step the Laplace transformation is applied to the reduced problem to obtain truncated solution. At the final step using the inverse transformations, we get the truncated solution of the problem we consider it. Presented examples illustrate the applicability and power of the approaches, used in this study

Numerical solution of fuzzy parabolic differential equations by a finite difference methods

In this study, we consider the concept of under generalized differentiability for the fuzzy parabolic differential equations. When the fuzzy derivative is considered as generalization of the H-derivative, for our case, the fuzziness is in the coefficents as well as initial and boundary conditions. We analysed and applied to numerically solve a fuzzy parabolic equation by finite difference method. The applicability of presented algorithm is illustrated by solving an examples of fuzzy partial differential equations.Publisher's Versio

Approximate Solution of an Unknown Coefficients in Parabolic Equation

Abstract: In this paper, an inverse parabolic equation is solved by using the homotopy analysis method (HAM) and the homotopy perturbation method (HPM). The approximation solution of this equation is calculated in the form of series which its components are computed easily. HPM is shown not always to generate a continuous family of solutions in terms of the homotopy parameter. By the convergence-control parameter this can however be prevented to occur in the HAM. Illustrative examples are presented to exhibit a comparison between the HAM and the HPM

On Fractional Newton-Type Method for Nonlinear Problems

The current manuscript is concerned with the development of the Newton–Raphson method, playing a significant role in mathematics and various other disciplines such as optimization, by using fractional derivatives and fractional Taylor series expansion. The development and modification of the Newton–Raphson method allow us to establish two new methods, which are called first- and second-order fractional Newton–Raphson (FNR) methods. We provide convergence analysis of first- and second-order fractional methods and give a general condition for the convergence of higher-order FNR. Finally, some illustrative examples are considered to confirm the accuracy and effectiveness of both methods