The hyperbolic PARTIAL DIFFERENTIAL equation (PDE) has important practical uses in science and engineering. This article provides an estimate for solving the Goursat problem in hyperbolic linear PDEs with VARIABLE COEFFICIENTS. The Goursat PDE is transformed into a second kind of linear Volterra in-tegral equation. A convergent algorithm that employs Taylor polynomials is created to generate a collocation solution, and the error using the maxi-mum norm is estimated. The paper includes numerical examples to prove the method’s effectiveness and precision.

In this paper, we introduce the definition of a new general integral transform which call it "A new general polynomial transform ". Also, we introduce properties, theorems, proofs and transforms of the Logarithmic functions, polynomials functions and other functions. As well as, we discuss how we can apply this integral transform and its inverse to solve the ordinary DIFFERENTIAL EQUATIONS with VARIABLE COEFFICIENTS.

In this paper, reproducing kernel method is presented to obtain the analytical and approximate solutions to system of ordinary DIFFERENTIAL EQUATIONS with VARIABLE COEFFICIENTS. The solution obtained by using the method takes the form of a convergent series with easily computable components. In the mean time, the approximate solution un(x) is obtained by the n-term intercept of the analytical solution and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

In this paper, we apply the DIFFERENTIAL transform (DT) method for finding approximate solution of the system of linear and nonlinear Volterra integro-DIFFERENTIAL EQUATIONS with VARIABLE COEFFICIENTS, especially of higher order. We also obtain an error bound for the approximate solution. Since, in this method the COEFFICIENTS of Taylor series expansion of solution is obtained by a recurrence relation, thus we can use arbitrary number of Taylor series terms to obtain solutions with desired accuracy. Here we give some preliminary results of the DIFFERENTIAL transform and show that the DT method can be easily applied to a wide class of linear and nonlinear systems. Finally, the accuracy and simplicity of this method will be verified by solving some examples.

In this paper, some meshless methods based on the local Newton basis functions are used to solve some time dependent PARTIAL DIFFERENTIAL EQUATIONS. For stability reasons, used variably scaled radial kernels for constructing Newton basis functions. In continuation, with considering presented basis functions as trial functions, approximated solution functions in the event of spatial VARIABLE with collocation method. Then, with aid of method of lines obtained a system of ordinary DIFFERENTIAL EQUATIONS according to solution function in the event of time. Methods applied for solving the nonlinear Burgers’ equation and couple Burgers’ equation. The numerical results show that the proposed method is efficient, accurate and stable.

We expand a new generalization of the two-dimensional DIFFERENTIAL trans-form method. The new generalization is based on the two-dimensional differ-ential transform method, fractional power series expansions, and conformable fractional derivative. We use the new method for solving a nonlinear con-formable fractional PARTIAL DIFFERENTIAL equation and a system of conformable fractional PARTIAL DIFFERENTIAL equation. Finally, numerical examples are pre-sented to illustrate the preciseness and effectiveness of the new technique.