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Research Papers

Solutions of Delayed Partial Differential Equations With Space-Time Varying Coefficients

[+] Author and Article Information
Venkatesh Deshmukh

 Department of Mechanical Engineering, 800 Lancaster Avenue, Villanova, PA 19085venkatesh.deshmukh@villanova.edu

J. Comput. Nonlinear Dynam 7(2), 021002 (Dec 22, 2011) (7 pages) doi:10.1115/1.4005081 History: Received February 26, 2011; Revised September 02, 2011; Published December 22, 2011; Online December 22, 2011

A constructive algorithm using Chebyshev spectral collocation is proposed for computing trustworthy approximate solutions of linear and weakly nonlinear delayed partial differential equations or initial boundary value problems, with continuous and bounded coefficients. The boundary conditions are assumed to be Dirichlet. The solution of linear problems is obtained at Chebyshev grid points in space and a given interval of time. The algorithm is then extended to systems with weak nonlinearities using perturbation series, which yields nonhomogeneous initial boundary value problems without delay. The proposed methodology is illustrated using examples of linear and weakly nonlinear heat and wave equations with bounded continuous space-time varying coefficients.

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Figures

Grahic Jump Location
Figure 1

Linear heat equation solution: ɛ=0, τ=0.24, u1=0

Grahic Jump Location
Figure 2

Linear wave equation solution: ɛ=0, τ=0.24, u2=0

Grahic Jump Location
Figure 3

Nonlinear heat equation solution: ɛ=0.2, τ=0.48, u1=0

Grahic Jump Location
Figure 4

Nonlinear wave equation solution: ɛ=0.2, τ=0.64, u2=0

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