## Abstract

In this work, we focus on the fractional versions of the well-known Kolmogorov forward equations. We consider the problem in two cases. In case 1, we apply the left Caputo fractional derivatives for α ∈ (0, 1] and in case 2, we use the right Riemann-Liouville fractional derivatives on R_{+}, for α ∈ (1, +∞). The exact solutions are obtained for the both cases by Laplace transforms and stable subordinators.

Fractional Differential Equations . Springer, 2010. [14] G uo , D., and L akshmikantham , V. Nonlinear Problems in Abstract Cones . Academic Press, 1988. [15] H ouas , M., and B ezziou , M. Existence and stability results for fractional differential equations with two caputo fractional derivatives. Facta. Univ. Ser. Math. Inform. 34 , 2 (2019), 341–357. [16] K halil , R., H orani , M. A., Y ousef , A., and S ababheh , M. A new definition of fractional derivative. J. Comput. Appl. Math. 264 (2014), 65–70. [17] K ilbas , A., S rivastava , H., and T rujillo , J

( a , b ) and n − 1 < α ≤ n , the Caputo fractional derivative operator of order α is given as a C D t α f ( t ) = 1 Γ ( n − 1 ) ∫ a t f ( n ) ( τ ) ( t − τ ) α + 1 − n d τ . \ _a^CD_t^\alpha f(t) = {1 \over {\Gamma (n - 1)}}\int_a^t {{{f^{(n)}}(\tau )} \over {{{(t - \tau )}^{\alpha + 1 - n}}}}\;d\tau . Throughout this paper, we denote the Caputo fractional derivative operator as a C D t α = D a α \ _a^CD_t^\alpha = D_a^\alpha . We also let a = 0 since our formulation only involves the initial conditions as t = 0. Definition 2 [ 20 ] The Riemann

fractional calculus have played a very important role in the various fields such as chemistry, biology, engineering, economics and signal processing. At this point, it should be pointed out that several definitions have been proposed of a fractional derivative, among those the Riemann - Liouville and Caputo fractional derivatives are the most popular. The differential equations defined in terms of Riemann - Liouville derivatives require fractional initial conditions, whereas the differential equations defined in terms of Caputo derivatives require regular boundary

## Abstract

This paper deals with the existence of solutions for a class of boundary value problem (BVP) of fractional differential equation with three point conditions via Leray-Schauder nonlinear alternative. Moreover, the existence of nonnegative solutions is discussed.

## Abstract

In this paper, we have extended the Fractional Differential Transform method for the numerical solution of the system of fractional partial differential-algebraic equations. The system of partial differential-algebraic equations of fractional order is solved by the Fractional Differential Transform method. The results exhibit that the proposed method is very effective.

## Abstract

In this paper we study existence and uniqueness of solutions for a coupled system consisting of fractional differential equations of Caputo type, subject to Riemann–Liouville fractional integral boundary conditions. The uniqueness of solutions is established by Banach contraction principle, while the existence of solutions is derived by Leray–Schauder’s alternative. We also study the Hyers–Ulam stability of mentioned system. At the end, examples are also presented which illustrate our results.

this method, obtained numerical results are very good and efficient for given examples. Definition 1 The Caputo fractional derivative D t α u ( t , x ) D_t^\alpha u\left( {t,x} \right) ( t,x ) of order α with respect to time is defined as: (2) ∂ α u ( t , x ) ∂ t α = D t α u ( t , x ) = 1 Γ ( n − α ) ∫ 0 t 1 ( t − p ) α − n + 1 ∂ α u ( p , x ) ∂ p α dp , ( n − 1 < α < n ) \matrix{ {{{{\partial ^\alpha }u(t,x)} \over {\partial {t^\alpha }}} = D_t^\alpha u(t,x)} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {1

## Abstract

In this paper, we propose a new approximate method, namely fractional natural decomposition method (FNDM) in order to solve a certain class of nonlinear time-fractional wave-like equations with variable coefficients. The fractional natural decomposition method is a combined form of the natural transform method and the Adomian decomposition method. The nonlinear term can easily be handled with the help of Adomian polynomials which is considered to be a clear advantage of this technique over the decomposition method. Some examples are given to illustrate the applicability and the easiness of this approach.

## Abstract

In this paper, we present numerical solution for the fractional Bratu-type equation via fractional residual power series method (FRPSM). The fractional derivatives are described in Caputo sense. The main advantage of the FRPSM in comparison with the existing methods is that the method solves the nonlinear problems without using linearization, discretization, perturbation or any other restriction. Three numerical examples are given and the results are numerically and graphically compared with the exact solutions. The solutions obtained by the proposed method are in complete agreement with the solutions available in the literature. The results reveal that the FRPSM is a very effective, simple and efficient technique to handle a wide range of fractional differential equations.