This work discusses the beginning of fractional calculus and how the Sumudu and Elzaki transforms are applied to fractional derivatives. This approach combines a double Sumudu-Elzaki transform strategy to discover analytic solutions to space-time fractional partial differential equations in Mittag-Leffler functions subject to initial and boundary conditions. Where this method gets closer and closer to the correct answer, and the technique's efficacy is demonstrated using numerical examples performed with Matlab R2015a.

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

In this work, we will combine the Laplace transform method with the Adomian decomposition method and modified Adomian decomposition method for semi-analytic treatments of the nonlinear integro-fractional differential equations of the Volterra-Hammerstein type with difference kernel and such a problem which the kernel has a first order simple degenerate kind which the higher-multi fractional derivative is described in the Caputo sense. In these methods, the solution of a functional equation is considered as the sum of infinite series of components after applying the inverse of Laplace transformation usually converging to the solution, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical

    In this work, a class of stochastically perturbed differential systems with standard Brownian motion of ordinary unperturbed differential system is considered and studied. The necessary conditions for the existence of a unique solution of the stochastic perturbed semi-linear system of differential equations are suggested and supported by concluding remarks. Some theoretical results concerning the mean square exponential stability of the nominal unperturbed deterministic differential system and its equivalent stochastically perturbed system with the deterministic and stochastic process as a random noise have been stated and proved. The proofs of the obtained results are based on using the stochastic quadratic Lyapunov function meth

 In this paper, we introduce and discuss an algorithm for the numerical solution of some kinds of fractional integral and fractional integrodifferential equations. The algorithm for the numerical solution of these equations is based on iterative approach. The stability and convergence of the fractional order numerical method are described. Finally, some numerical examples are provided to show that the numerical method for solving the fractional integral and fractional integrodifferential equations is an effective solution method.

In this article, the backstepping control scheme is proposed to stabilize the fractional order Riccati matrix differential equation with retarded arguments in which the fractional derivative is presented using Caputo's definition of fractional derivative. The results are established using Mittag-Leffler stability. The fractional Lyapunov function is defined at each stage and the negativity of an overall fractional Lyapunov function is ensured by the proper selection of the control law. Numerical simulation has been used to demonstrate the effectiveness of the proposed control scheme for stabilizing such type of Riccati matrix differential equations.

In this article, an efficient reliable method, which is the residual power series method (RPSM), is used in order to investigate the approximate solutions of conformable time fractional nonlinear evolution equations with conformable derivatives under initial conditions. In particular, two types of equations are considered, which are time coupled diffusion-reaction equations (CD-REs) and MKdv equations coupled with conformable fractional time derivative of order α. The attitude of RPSM and the influence of different values of α are shown graphically.

In this paper, we discuss the difference between classical and nonclassical symmetries. In addition, we found the non-classical symmetry of the Benjamin Bona Mahony Equation (BBM). Finally, we found a new exact solution to a Benjamin Bona Mahony Equation (BBM) using nonclassical symmetry.

In this paper, we present an approximate analytical and numerical solutions for the differential equations with multiple delay using the extend differential transform method (DTM). This method is used to solve many linear and non linear problems.

In this paper, the series solution is applied to solve third order fuzzy differential equations with a fuzzy initial value. The proposed method applies Taylor expansion in solving the system and the approximate solution of the problem which is calculated in the form of a rapid convergent series; some definitions and theorems are reviewed as a basis in solving fuzzy differential equations. An example is applied to illustrate the proposed technical accuracy. Also, a comparison between the obtained results is made, in addition to the application of the crisp solution, when the-level equals one.