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Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on studying a general form of pantograph type Volterra integro-differential equations (PVIDEs). We apply a new collocation spectral approach, based on shifted Chebyshev polynomials, for converting such PVIDEs into systems of algebraic equations. In addition, we apply the new spectral approach for systems of pantograph type Volterra integro-differential equations (SPVIDEs). We investigate the error analysis of the proposed numerical approach. Also, we present some comparisons with other spectral approaches for clarifying the superiority of the new spectral approach. [ABSTRACT FROM AUTHOR]

Spectral methods for solving differential/integral equations are characterized by the representation of the solution by a truncated series of smooth functions. The unknowns to be determined are the expansion coefficients in such a representation. The goal of this article is to give an overview of numerical problems encountered when determining these coefficients and the rich variety of techniques proposed to solve these problems. Therefore, a series of explicit formulae expressing the derivatives, integrals and moments of a class of orthogonal polynomials of any degree and for any order in terms of the same polynomials are addressed. We restrict the current study to the orthogonal polynomials of the Hermite, generalized Laguerre, Bessel, and Jacobi (including Legendre, Chebyshev, and ultraspherical) families. Moreover, formulae expressing the coefficients of an expansion of these polynomials which have been differentiated or integrated an arbitrary number of times in terms of the coefficients of the original expansion are given. In addition, formulae for the polynomial coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients are also presented. A simple approach to build and solve recursively for the connection coefficients between different orthogonal polynomials is established. The essential results are summarized in tables which could serve as a useful reference to numerical analysts and practitioners. Finally, applications of these results in solving differential and integral equations with varying polynomial coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, are implemented. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient spectral numerical method for solving sine and Klein-Gordon nonlinear variable-order fractional differential equations with the initial and Dirichlet boundary conditions. The approach is based on the shifted Legendre-Gauss and Chebyshev-Gauss collocation methods. The Caputo fractional derivative of variable order is adopted, and the original problems are reduced to systems of algebraic equations. The validity and effectiveness of the method is demonstrated by means of several numerical examples. [ABSTRACT FROM AUTHOR]

This paper applies the shifted Jacobi-Gauss collocation (SJ-G-C) method for solving variable-order fractional integro-differential equations (VO-FIDE) with initial conditions. The Riemann-Liouville fractional derivative, Dν(x), and integral, Iν(x), of variable order are combined, and the SJ-G-C applied to produce a system of algebraic equations. Numerical experiments demonstrate the applicability and reliability of the algorithm when compared with current methods. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to implementing spectral numerical solutions to two kinds of fractional space-time advection-dispersion problems governed by certain constraints conditions. The collocation and tau spectral methods are utilized for obtaining the proposed spectral solutions. A double Legendre expansion is proposed as an approximate solution. The main idea of the algorithm is basically depend on converting the equation with its constraints conditions into linear or nonlinear systems of algebraic equations which can be efficiently solved with the aid of suitable numerical solvers. Some illustrative examples are displayed aiming to confirm robustness, efficiency and accuracy of the proposed spectral solutions. [ABSTRACT FROM AUTHOR]

This article presents some spectral Petrov-Galerkin numerical algorithms based on using Chebyshev polynomials of third and fourth kinds for solving the integrated forms of high odd-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions. The principle idea behind obtaining the proposed numerical algorithms is based on constructing trial and test functions as compact combinations of shifted Chebyshev polynomials of third and fourth kinds. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Some numerical examples are illustrated for the sake of demonstrating the validity and the applicability of the proposed algorithms. The presented numerical results indicate that the proposed algorithms are reliable and very efficient. [ABSTRACT FROM AUTHOR]

In this paper, we propose the shifted Jacobi-Gauss collocation spectral method for solving initial value problems of Bratu type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The spatial approximation is based on shifted Jacobi polynomials Jn(α,β)( x) with α, β ∈ (−1, ∞), x ∈ [0, 1] and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes. Illustrative examples have been discussed to demonstrate the validity and applicability of the proposed technique. Comparing the numerical results of the proposed method with some well-known results show that the method is efficient and gives excellent numerical results. [ABSTRACT FROM AUTHOR]

Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultra-spherical polynomials are considered. Also the results for Chebyshev polynomials of the first, second, third and fourth kinds and of Legendre polynomials are noted. An application of how to use double Jacobi polynomials for solving Poisson's equation in two variables subject to nonhomogeneous mixed boundary conditions is described. [ABSTRACT FROM AUTHOR]