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Research Papers

# An Accurate Jacobi Pseudospectral Algorithm for Parabolic Partial Differential Equations With Nonlocal Boundary Conditions

[+] Author and Article Information
E. H. Doha

Department of Mathematics,
Faculty of Science,
Cairo University,
Giza 12613, Egypt
e-mail: eiddoha@frcu.eun.eg

A. H. Bhrawy

Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia
Department of Mathematics,
Faculty of Science,
Beni-Suef University,
Beni-Suef 62511, Egypt
e-mail: alibhrawy@yahoo.co.uk

M. A. Abdelkawy

Department of Mathematics,
Faculty of Science,
Beni-Suef University,
Beni-Suef 62511, Egypt
e-mail: melkawy@yahoo.com

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received December 30, 2013; final manuscript received February 20, 2014; published online January 12, 2015. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 10(2), 021016 (Mar 01, 2015) (13 pages) Paper No: CND-13-1330; doi: 10.1115/1.4026930 History: Received December 30, 2013; Revised February 20, 2014; Online January 12, 2015

## Abstract

A new spectral Jacobi–Gauss–Lobatto collocation (J–GL–C) method is developed and analyzed to solve numerically parabolic partial differential equations (PPDEs) subject to initial and nonlocal boundary conditions. The method depends basically on the fact that an expansion in a series of Jacobi polynomials $Jn(θ,ϑ)(x)$ is assumed, for the function and its space derivatives occurring in the partial differential equation (PDE), the expansion coefficients are then determined by reducing the PDE with its boundary conditions into a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit the Runge–Kutta (IRK) method of order four. The proposed method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the spatial discretizations. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.

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## Figures

Fig. 1

The numerical solution u˜(x,t) of the problem in Eq. (4.1) where θ = ϑ = 0 and N = 12

Fig. 2

The absolute error of the problem in Eq. (4.1), where θ = ϑ = 0 and N = 12

Fig. 3

The curves of numerical and exact solutions of the problem in Eq. (4.1) for the different values of t = 0.5, 1.0, and 1.5 where θ = ϑ = 0 and N = 12

Fig. 4

The curves of numerical and exact solutions of the problem in Eq. (4.1) for the different values of x = 0.5, 1.0, and 1.5 where θ = ϑ = 0 and N = 12

Fig. 5

The curve of absolute error of the problem in Eq. (4.1) at x = 0, where θ = ϑ = 0 and N = 12

Fig. 6

The numerical solution u˜(x,t) of the problem in Eq. (4.7), where -θ = ϑ = 1/2 and N = 10

Fig. 7

The curves of numerical and exact solutions of the problem in Eq. (4.7) for the different values of t = 0.1, 0.5, and 0.9 where -θ = ϑ = 1/2 and N = 10

Fig. 8

The curves of numerical and exact solutions of the problem in Eq. (4.7) for the different values of x = 0.1, 0.2, and 0.3 where θ = ϑ = 0,ɛ = 2-4 and N = 16

Fig. 9

The absolute error of the problem in Eq. (4.7), at x = 0, where -θ = ϑ = 1/2 and N = 10

Fig. 10

The absolute error of the problem in Eq. (4.7) at x = 0, where -θ = ϑ = 1/2 and N = 10

Fig. 11

The numerical solution u˜(x,t) of the problem in Eq. (4.12), where θ = ϑ = -1/2 and N = 8

Fig. 12

The absolute error of the problem in Eq. (4.12), where θ = ϑ = -1/2 and N = 8

Fig. 13

The curves of numerical and exact solutions of the problem in Eq. (4.12) for the different values of t = 1.0, 1.5, and 2.0 where θ = ϑ = -1/2 and N = 8

Fig. 14

The curves of numerical and exact solutions of the problem in Eq. (4.12) for the different values of x = 0.1, 0.2, and 0.3 where θ = ϑ = -1/2 and N = 8

Fig. 15

The absolute error of the problem in Eq. (4.12) at x = 0, where θ = ϑ = -1/2 and N = 8

Fig. 16

The absolute error of the problem in Eq. (4.12) at t = 0, where θ = ϑ = -1/2 and N = 8

Fig. 17

The numerical solution u˜(x,t) of the problem in Eq. (4.15), where θ = ϑ = 1/2 and N = 8

Fig. 18

The absolute error of the problem in Eq. (4.15), where θ = ϑ = 1/2 and N = 8

Fig. 19

The curves of numerical and exact solutions of the problem in Eq. (4.15) for the different values of t = 0.1, 0.2, and 0.3 where θ = ϑ = 1/2 and N = 8

Fig. 20

The curves of numerical and exact solutions of the problem in Eq. (4.15) for the different values of t = 0.1, 0.2, and 0.3 where θ = ϑ = 1/2 and N = 8

Fig. 21

The numerical solution u˜(x,t) of the problem in Eq. (4.18), where θ = ϑ = -1/2 and N = 10

Fig. 22

The curves of numerical and exact solutions of the problem in Eq. (4.18) for the different values of t = 0.1, 0.5, and 1.0 where θ = ϑ = -1/2 and N = 10

Fig. 23

The curves of numerical and exact solutions of the problem in Eq. (4.18) for the different values of x = 0.1, 0.2, and 0.3 where θ = ϑ = -1/2 and N = 10

Fig. 24

The absolute error of the problem in Eq. (4.18) at x = 0, where θ = ϑ = -1/2 and N = 10

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