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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]

This article, presented a shifted Legendre Gauss‐Lobatto collocation (SL‐GL‐C) method which is introduced for solving variable‐order fractional Volterra integro‐differential equation (VO‐FVIDEs) subject to initial or nonlocal conditions. Based on shifted Legendre Gauss‐Lobatto (SL‐GL) quadrature, we treat with integral term in the aforementioned problems. Via the current approach, we convert such problem into a system of algebraic equations. After that we obtain the spectral solution directly for the proposed problem. The high accuracy of the method was proved by several illustrative examples. [ABSTRACT FROM AUTHOR]

The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes an integer derivative in addition to the fractional derivative. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Based on the shifted Legendre orthonormal polynomials, we employ the operational matrix of fractional derivatives, the Legendre–Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations. The convergence of the proposed method is analyzed. For confirming the validity and accuracy of the proposed numerical method, a numerical example is presented along with a comparison between our numerical results and those obtained using the Legendre spectral-collocation method. [ABSTRACT FROM AUTHOR]

The telegraph equation describes various phenomena in many applied sciences. We propose two new efficient spectral algorithms for handling this equation. The principal idea behind these algorithms is to convert the linear/nonlinear telegraph problems (with their initial and boundary conditions) into a system of linear/nonlinear equations in the expansion coefficients, which can be efficiently solved. The main advantage of our algorithm in the linear case is that the resulting linear systems have special structures that reduce the computational effort required for solving them. The numerical algorithms are supported by a careful convergence analysis for the suggested Chebyshev expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithms. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1553-1571, 2016 [ABSTRACT FROM AUTHOR]

In this paper, a shifted Jacobi-Gauss collocation spectral algorithm is developed for solving numerically systems of high-order linear retarded and advanced differential-difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi-Gauss interpolation nodes as collocation nodes. The system of differential-difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient. [ABSTRACT FROM AUTHOR]

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small. [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]

We solve three versions of nonlinear time-dependent Burgers-type equations. The Jacobi-Gauss-Lobatto points are used as collocation nodes for spatial derivatives. This approach has the advantage of obtaining the solution in terms of the Jacobi parameters α and β. In addition, the problem is reduced to the solution of the system of ordinary differential equations (SODEs) in time. This system may be solved by any standard numerical techniques. Numerical solutions obtained by this methodwhen compared with the exact solutions reveal that the obtained solutions produce high-accurate results. Numerical results show that the proposed method is of high accuracy and is efficient to solve the Burgers-type equation. Also the results demonstrate that the proposed method is a powerful algorithm to solve the nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear second-order two-point boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinearmultipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourthkind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably with the analytical known solutions. [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]

This paper presents some efficient spectral algorithms for solving linear sixth-order two-point boundary value problems in one dimension based on the application of the Galerkin method. The proposed algorithms are extended to solve the two-dimensional sixth-order differential equations. A family of symmetric generalized Jacobi polynomials is introduced and used as basic functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. The various matrix systems resulting from the proposed algorithms are carefully investigated, especially their condition numbers and their complexities. These algorithms are extensions to some of the algorithms proposed by Doha and Abd-Elhameed (2002) and Doha and Bhrawy (2008) for second- and fourth-order elliptic equations, respectively. Three numerical results are presented to demonstrate the efficiency and the applicability of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A Jacobi dual-Petrov-Galerkin (JDPG) method is introduced and used for solving fully integrated reformulations of third- and fifth-order ordinary differential equations (ODEs) with constant coefficients. The reformulated equation for the Jth order ODE involves n-fold indefinite integrals for n = 1,…, J. Extension of the JDPG for ODEs with polynomial coefficients is treated using the Jacobi-Gauss-Lobatto quadrature. Numerical results with comparisons are given to confirm the reliability of the proposed method for some constant and polynomial coefficients ODEs. [ABSTRACT FROM AUTHOR]

Two new formulae expressing explicitly the integrals of Laguerre (Hermite) polynomials of any degree and for any order in terms of the Laguerre (Hermite) polynomials themselves are proved. Another two new explicit formulae relating the Laguerre (Hermite) coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function are also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed. [ABSTRACT FROM AUTHOR]

A formula expressing explicitly the difference derivatives of Hahn polynomials of any degree and for any order in terms of Hahn polynomials themselves is proved. Another explicit formula, which expresses the Hahn expansion coefficients of a general-order difference derivative of an arbitrary polynomial of a discrete variable in terms of its original Hahn coefficients, is also given. A formula for the Hahn coefficients of the moments of one single Hahn polynomial of certain degree is proved. A formula for the Hahn coefficients of the moments of a general-order difference derivative of an arbitrary polynomial of a discrete variable in terms of its Hahn coefficients is also obtained. Application of these formulae for solving ordinary difference equations with varying polynomial coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Hahn-Hahn, Meixner-Hahn, Kravchuk-Hahn and Charlier-Hahn is also developed. [ABSTRACT FROM AUTHOR]

Two formulae expressing explicitly the difference derivatives and the moments of a discrete orthogonal polynomials { P n ( x ): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n ( x ) themselves are proved. Two other formulae for the expansion coefficients of a general-order difference derivatives ∇ q f ( x ), and for the moments x ℓ ∇ q f ( x ), of an arbitrary function f ( x ) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica), in order to build and solve recursively for the connection coefficients between two families of Meixner, Kravchuk and Charlier, is described. Three analytical formulae for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are also developed. [ABSTRACT FROM AUTHOR]

A formula expressing the Hermite coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Hermite polynomials of any degree and for any order as a linear combination of suitable Hermite polynomials is deduced. A formula for the Hermite coefficients of the moments of one single Hermite polynomial of certain degree is given. Formulae for the Hermite coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Hermite coefficients are also obtained. Two numerical applications of how to use these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in Hermite coefficients, are discussed. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi-Hermite and Laguerre-Hermite polynomials is described. Explicit formula for these coefficients between Jacobi and Hermite polynomials is given, of which the ultraspherical polynomials and the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Analytical formula for the connection coefficients between Laguerre and Hermite polynomials is also obtained. [ABSTRACT FROM AUTHOR]

In this manuscript, we report a new operational technique for approximating the numerical solution of fractional optimal control (FOC) problems. The operational matrix of the Caputo fractional derivative of the orthonormal Chebyshev polynomial and the Legendre–Gauss quadrature formula are used, and then the Lagrange multiplier scheme is employed for reducing such problems into those consisting of systems of easily solvable algebraic equations. We compare the approximate solutions achieved using our approach with the exact solutions and with those presented in other techniques and we show the accuracy and applicability of the new numerical approach, through two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present an ultraspherical wavelets-Gauss collocation method for obtaining direct solutions of first- and second-order nonlinear differential equations subject to homogenous and nonhomogeneous initial conditions. The properties of ultraspherical wavelets are used to reduce the differential equations with their initial conditions to systems of algebraic equations, which then must be solved by using suitable numerical solvers. The function approximations are spectral and have been chosen in such a way that make them easy to calculate the expansion coefficients of the thought-for solutions. Uniqueness and convergence of the proposed function approximation is discussed. Four illustrative numerical examples are considered and these results are comparing favorably with the analytic solutions and proving more accurate than those discussed by some other existing techniques in the literature. [ABSTRACT FROM AUTHOR]