Numerical methods have essential role to approximate the solutions of Partial Differential Equations (PDEs). Spectral method is one of the best numerical methods of exponential order with high convergence rate to solve PDEs. In recent decades the Chebyshev Spectral collocation (CSC) method has been used to approximate solutions of linear PDEs. In this paper, by using linear algebra operators, we implement Kronecker Chebyshev Spectral collocation (KCSC) method for n-order linear PDEs. By statistical tools, we obtain that the Run times of KCSC method has polynomial growth, but the Run times of CSC method has exponential growth. Moreover, error upper bounds of KCSC and CSC methods are compared.

The recognition and the calculation of all branches of solutions of the nonlinear boundary value problems is di cult obviously. The complexity of this issue goes back to the being nonlinearity of the problem. Regarding this matter, this paper considers steady state reactive transport model which does not have exact closed{form solution and discovers existence of dual or triple solutions in some cases using a new hybrid method based on pseudo{Spectral collocation in the sense of least square method. Furthermore, the method usages Picard iteration and Newton method to treat nonlinear term in order to obtain unique and multiple solutions of the problem, respectively.

In this paper, we present a numerical method to approximate the solution of the multi-term time fractional diffusion-wave equation (M-TFDWE). The proposed method represents the solution as a sum of shifted Gegenbauer polynomials (SGPs) with unknown coefficients. By using the operational matrix of fractional integration and integer derivatives based on SGPs, the M-TFDWE is converted into a system of algebraic equations. The convergence analysis of this numerical method is also discussed. Finally, we provide two examples to illustrate the accuracy of the proposed method.

In this paper, the Chebychev Spectral collocation method is applied for the thermal analysis ofconvective-radiative straight fins with the temperature-dependent thermal conductivity. The developed heat transfermodel was used to analyse the thermal performance, establish the optimum thermal design parameters, and also, investigate the effects of thermo-geometric parameters and thermal conductivity (nonlinear) parameters on thethermal performance of the fin. The results of this study reveal that the rate of heat transfer from the fin increases asthe convective, radioactive, and magnetic parameters increase. This study establishes good agreement between theobtained results using Chebychev Spectral collocation method and the results obtained using Runge-Kutta methodalong with shooting, homotopy perturbation, and adomian decomposition methods.

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.