In this work, we are concerned with how to find an explicit approximate solution (AS) for the telegraph equation of space-fractional order (TESFO) using Sumudu transform method (STM). In this method, the space-fractional order derivatives are defined in the Caputo idea. The Sumudu method (SM) is established to be reliable and accurate. Three examples are discussed to check the applicability and the simplicity of this method. Finally, the Numerical results are tabulated and displayed graphically whenever possible to make comparisons between the AS and exact solution (ES).

The approach given in this paper leads to numerical methods to find the approximate solution of volterra integro –diff. equ.1st kind. First, we reduce it from integro VIDEs to integral VIEs of the 2nd kind by using the reducing theory, then we use two types of Non-polynomial spline function (linear, and quadratic). Finally, programs for each method are written in MATLAB language and a comparison between these two types of Non-polynomial spline function is made depending on the least square errors and running time. Some test examples and the exact solution are also given.

A non-polynomial spline (NPS) is an approximation method that relies on the triangular and polynomial parts, so the method has infinite derivatives of the triangular part of the NPS to compensate for the loss of smoothness inherited by the polynomial. In this paper, we propose polynomial-free linear and quadratic spline types to solve fuzzy Volterra integral equations (FVIE) of the 2nd kind with the weakly singular kernel (FVIEWSK) and Abel's type kernel. The linear type algorithm gives four parameters to form a linear spline. In comparison, the quadratic type algorithm gives five parameters to create a quadratic spline, which is more of a credit for the exact solution. These algorithms process kernel singularities with a simple techniqu

In this paper we use non-polynomial spline functions to develop numerical methods to approximate the solution of 2nd kind Volterra integral equations. Numerical examples are presented to illustrate the applications of these method, and to compare the computed results with other known methods.

The aim of this paper is adopted to give an approximate solution for advection dispersion equation of time fractional order derivative by using the Chebyshev wavelets-Galerkin Method . The Chebyshev wavelet and Galerkin method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are described based on the Caputo sense. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

In this paper, third order non-polynomial spline function is used to solve 2nd kind Volterra integral equations. Numerical examples are presented to illustrate the applications of this method, and to compare the computed results with other known methods.

In this paper, author’s study sub diffusion bio heat transfer model and developed explicit finite difference scheme for time fractional sub diffusion bio heat transfer equation by using caputo fabrizio fractional derivative. Also discussed conditional stability and convergence of developed scheme. Furthermore numerical solution of time fractional sub diffusion bio heat transfer equation is obtained and it is represented graphically by Python.

      In this paper we shall prepare an  sacrificial solution for fuzzy differential algebraic equations of fractional order (FFDAEs) based on the Adomian decomposition method (ADM) which is proposed to solve (FFDAEs) . The blurriness will appear in the boundary conditions, to be fuzzy numbers. The solution of the proposed pattern of  equations is studied in the form of a convergent series with readily computable components. Several examples are resolved as  clarifications, the numerical outcomes are obvious that the followed approach is simple to perform and precise when utilized to (FFDAEs).

      In this paper we shall prepare an  sacrificial solution for fuzzy differential algebraic equations of fractional order (FFDAEs) based on the Adomian decomposition method (ADM) which is proposed to solve (FFDAEs) . The blurriness will appear in the boundary conditions, to be fuzzy numbers. The solution of the proposed pattern of  equations is studied in the form of a convergent series with readily computable components. Several examples are resolved as  clarifications, the numerical outcomes are obvious that the followed approach is simple to perform and precise when utilized to (FFDAEs).