H.M. El-Hawary

A Chebyshev approximation for solving optimal control problems

Abstract This paper presents a numerical solution for solving optimal control problems, and the controlled Duffing oscillator. A new Chebyshev spectral procedure is introduced. Control variables and state variables are approximated by Chebyshev series. Then the system dynamics is transformed into systems of algebraic equations. The optimal control problem is reduced to a constrained optimization problem. Results and comparisons are given at the end of the paper.

Chebyshev solution of laminar boundary layer flow

An expansion procedure using the Chebyshev polynomials is proposed by using El-Gendi method [1], which yields more accurate results than those computed by P. M. Beckett [2] and A. R. Wadia and F. R. Payne [6] as indicated from solving the Falkner-Skan equation, which uses a boundary value technique. This method is accomplished by starting with Chebyshev approximation for the highest-order derivative and generating approximations to the lower-order derivatives through integration of the highest-order derivative.

An optimal ultraspherical approximation of integrals

In this paper some useful properties of the ultraspherical polynomials are derived. An ultraspherical approximation of any continuous function is presented. An ultraspherical approximation of finite integrals is introduced. The error estimation for the ultraspherical approximation is derived. An algorithm that gives an optimal approximation of the integrals is introduced. Numerical results are included to confirm the efficiency and accuracy of the method.

On some 4-point spline collocation methods for solving second-order initial value problems

Abstract This paper discusses collocation schemes based on seventh C 3 -splines with four collocation points x i−1+α =x i−1 +αh,x i−1+β =x i−1 +βh,x i−1+θ =x i−1 +θh and x i =x i−1 +h in each subinterval [x i−1 ,x i ], i=1(1)N , for solving second-order initial value problems in ordinary differential equations including stiff equations. Here 0 are arbitrarily given. It is shown that the methods are convergent and the order of convergence is seven if: 1−α−β−θ+αβ+αθ+βθ−2αβθ⩽0 and they are unstable if α,β,θ . Moreover, the absolute stability properties of the methods are considered. It shows that with 0.888035⩽α the methods possess unbounded regions of absolute stability, on other hand, the sizes of regions of absolute stability increase when α,β and θ are close to 1 − .

A high-order nodal discontinuous Galerkin method for a linearized fractional CahnHilliard equation

In this paper, we develop and analyze a nodal discontinuous Galerkin method for the linearized fractional CahnHilliard equation containing derivatives of fractional order in space. The Caputo derivative is chosen as the representation of spatial derivative. The linearized fractional CahnHilliard problem has been expressed as a system of low order differential/integral equations. We adopt the nodal discontinuous Galerkin methods for the full spatial discretization using a high-order nodal basis set of orthonormal LagrangeLegendre polynomials of arbitrary order in space on each element of computational domain. Moreover, we prove the stability and optimal order of convergence N+1 for the linearized fractional CahnHilliard problem when polynomials of degree N are used. Numerical experiments are displayed to verify the theoretical results.

Chebyshev solution of axisymmetric stagnation flow on a cylinder

A numerical procedure, based on Chebyshev polynomials, has been developed for a class of problems in the boundary layer theory of a micropolar fluid. Here, the results are presented for the steady micropolar fluid in the vicinity of an axisymmetric stagnation point flow on a cylinder. This method is accomplished by starting with a Chebyshev approximation for the highest order derivatives and generating approximations to the lower order derivatives. The numerical solutions are presented for a range of values of material parameters and Reynolds number. Furthermore, the results have been compared with the results of the corresponding flow of a Newtonian fluid. This type of stagnation flow has applications in certain cooling processes.

Legendre spectral method for solving integral and integro-differential equations

A new spectral approximation of an integral based on Legendre approximation at the zeros of the first term of the residual is presented. The method is used to solve integral and integro-differential equations. The method generates approximations to the lower order derivatives of the function through successive integrations of the Legendre polynomial approximation to the highest order derivative. Numerical results are included to confirm the efficiency and accuracy of the method.

Similarity Analysis for Effects of Variable Diffusivity and Heat Generation/Absorption on Heat and Mass Transfer for a MHD Stagnation-Point Flow of a Convective Viscoelastic Fluid over a Stretching Sheet with a Slip Velocity

A mathematical analysis has been carried out for stagnation-point heat and mass transfer of a viscoelastic fluid over a stretching sheet with surface slip velocity, concentration dependent diffusivity, thermal convective boundary conditions, and heat source/sink. The governing partial differential equations are reduced to a system of nonlinear ordinary differential equations using Lie group analysis. Numerical solutions of the resulting ordinary differential equations are obtained using shooting method. The influences of various parameters on velocity, temperature, and mass profiles have been studied. Also, the effects of various parameters on the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number are given in graphics form and discussed.

On some 4-Point Spline Collocation Methods for Solving Ordinary Initial Value Problems

This paper discusses collocation schemes based on seventh C 3 -splines with four collocation points s_{i-1+\alpha}=x_{i-1}+\alpha h , x_{i-1+\beta}=x_{i-1}+\beta h , x_{i-1+\theta}=x_{i-1}+\theta h and x_i=x_{i-1}+h in each subinterval [ x i @ 1 , x i ], i = 1(1) N for solving initial value problems in ordinary differential equations including stiff equations. Here 0 < f < g < è < 1 are arbitrarily given. It is shown that the methods are convergent and the order of convergence is seven if: \eqalign {&\beta(\theta -\theta^2)+\beta^2(\theta^2-\theta)+\alpha^2[\theta^2-\theta +\beta^2(1+4\theta -5\theta^2)+\theta(4\theta^2-4\theta -1)]\cr &\qquad \qquad \qquad +\alpha[\theta -\theta^2+\beta (1+4\theta -4\theta^2)+\beta^2(4\theta^2+4\theta -1)]\le 0 and they are unstable if f , g , è < 0.7279115. Moreover, the absolute stability properties of the methods are considered. It shows that with 0.888035 h f < g < è < 1 the methods are A-stable while they are not if f h 0.5; on other hand, the sizes of regions of absolute stability increase when f , g , è M 1 m.

Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain

ABSTRACT In this article, we first introduce a singular fractional Sturm–Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.