In this paper, we introduce and discuss an algorithm for the numerical solution of two- dimensional fractional partial differential equation with parameter. The algorithm for the numerical solution of this equation is based on implicit and an explicit difference method. Finally, numerical example is provided to illustrate that the numerical method for solving this equation is an effective solution method.

This study focuses on studying an oscillation of a second-order delay differential equation. Start work, the equation is introduced here with adequate provisions. All the previous is braced by theorems and examplesthat interpret the applicability and the firmness of the acquired provisions

In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal meth

In this article, the solvability of some proposal types of the multi-fractional integro-partial differential system has been discussed in details by using the concept of abstract Cauchy problem and certain semigroup operators and some necessary and sufficient conditions. 

       In this work, Elzaki transform (ET) introduced by Tarig Elzaki is applied to solve linear Volterra fractional integro-differential equations (LVFIDE). The fractional derivative is considered in the Riemman-Liouville sense. The procedure is based on the application of (ET) to (LVFIDE) and using properties of (ET) and its inverse. Finally, some examples are solved to show that this is computationally efficient and accurate.

In this work, Elzaki transform (ET) introduced by Tarig Elzaki is applied to solve linear Volterra fractional integro-differential equations (LVFIDE). The fractional derivative is considered in the Riemman-Liouville sense. The procedure is based on the application of (ET) to (LVFIDE) and using properties of (ET) and its inverse. Finally, some examples are solved to show that this is computationally efficient and accurate.

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.

     Fractional calculus has paid much attention in recent years, because it plays an essential role in many fields of science and  engineering, where the study of stability theory of fractional differential equations emerges to be very important. In this paper, the stability of fractional order ordinary differential equations will be studied and introduced the backstepping method. The Lyapunov function  is easily found by this method. This method also gives a guarantee of stable solutions for the fractional order differential equations. Furthermore it gives asymptotically stable.

This paper aims to study the fractional differential systems arising in warm plasma, which exhibits traveling wave-type solutions. Time-fractional Korteweg-De Vries (KdV) and time-fractional Kawahara equations are used to analyze cold collision-free plasma, which exhibits magnet-acoustic waves and shock wave formation respectively. The decomposition method is used to solve the proposed equations. Also, the convergence and uniqueness of the obtained solution are discussed. To illuminate the effectiveness of the presented method, the solutions of these equations are obtained and compared with the exact solution. Furthermore, solutions are obtained for different values of time-fractional order and represented graphically.

The aim of this paper is to employ the fractional shifted Legendre polynomials (FSLPs) in the matrix form to approximate the fractional derivatives and find the numerical solutions of the one-dimensional space-fractional bioheat equation (SFBHE). The Caputo formula was utilized to approximate the fractional derivative. The proposed methodology applied for two examples showed its usefulness and efficiency. The numerical results showed that the utilized technique is very efficacious with high accuracy and good convergence.