In this paper‎, ‎a numerical scheme based on shifted Vieta-Lucas polynomials is utilised to solve mentioned equation‎. ‎The main characteristic of the presented method is to approximate Brownian motion with help of the Gauss-Legendre quadrature‎, ‎which makes calculations easier‎. ‎Another characteristic of this method are employed suitable collocation points to convert the stochastic equation under the study into a system of algebraic equations by using the operational matrices‎. ‎So that‎, ‎Newton's method is applied to solve them‎. The convergence analysis and error bound of the suggested method are well established‎. ‎Additionally‎, ‎the proofs related to the existence and uniqueness of the solutions for the equations under investigation have been provided‎. ‎In order to illustrate the effectiveness‎, ‎compatibility and plausibility of the proposed technique‎, ‎four numerical examples are presented.

A numerical method for the variable-order fractional functional differential equations (VO-FFDEs) has been developed. This method is based on approximation with shifted Legendre polynomials. The properties of the latter were stated, first. These properties, together with the shifted Gauss-Legendre nodes were then utilized to reduce the VO-FFDEs into a solution of matrix equation. Sequentially, the error estimation of the proposed method was investigated. The validity and efficiency of our method were examined and verified via numerical examples.

‎The main goal of this research work is to provide a numerical technique based on choosing a set of basis functions for handling the third-order time-fractional Korteweg–De Vries Burgers' equation‎. ‎The trial functions are selected for the shifted second-kind Chebyshev polynomials (S2KCPs) compatible with the problem's governing initial and boundary conditions‎. ‎The Spectral tau method transforms the equation and its underlying conditions into a nonlinear system of algebraic equations that can be efficiently numerically inverted with the standard Newton's iterative procedures after the approximate solutions have been expressed as a double expansion of the two chosen basis functions‎. ‎The truncation error is estimated‎. ‎Various numerical examples are displayed together with comparisons to other approaches in the literature to show the applicability and accuracy of the provided methodology‎. ‎Different numerical models are displayed and compared to other methods in the literature‎.

An efficient collocation method based on the shifted Legendre polynomials is implemented for solv-ing the Magnetohydrodynamic Hiemenz flow with variable wall temperature in a porous medium.In the presented method the need for guessing and correcting the initial values during the solution procedure is eliminated and by using the given boundary conditions of the problem a stable solution can be derived. Numerical results show influence of the Prandtl number, permeability parameter, Hartmann number and suction/blowing parameter on the velocity and temperature profiles. The skin friction coefficient and the rate of heat transfer given by the Spectral collocation method are in good agreement with those of the previous studies.