Differential equations are used to represent different scientific problems are handled efficiently by integral transformations, where integral transforms represent an easy and effective tool for solving many problems in the mentioned fields. This work utilizes the integral transform of the Complex SEE integral transformation to provide an efficient solution method for the difference and differential-difference equations by benefiting from the properties of this complex transform to solve some problems related to difference and differential-difference equations. The 3D, contour and 2D surfaces, as well as the related density plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of differential-difference problems with initial conditions.

In this paper, we solve the quadratic -functional equations 2f(x) + 2f(y) = f(x + y) +  􀀀 2f( (x 􀀀 y)); (0. 1) where  is a  xed non-Archimedean number with  􀀀 2 6= 3. Using the  xed point method and the direct method, we prove the Hyers-Ulam stability of the functional equation (0. 1) in non-Archimedean Banach spaces.

In this paper, the continuse Legendre wavelets constructed on the interval [0, 1] are used to solve the nonlinear Fredholm integro-differential equation. The nonlinear part of integro-differential is ap- proximated by Legendre wavelets, and the nonlinear integro-differential is reduced to a system of nonlinear equations. We give some numerical examples to show applicability of the proposed method.