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Theory and computation in hyperbolic model problems
Abstract
A Sinc-Galerkin procedure is developed for a system of partial differential equation of the form$$U\sb{t}(x,t)+F(U(x,t))\sb{x}=H(x,t,U(x,t)),\ (x,t) \in \IR\times (0,T)$$with the initial condition$$U(x,0)=U\sp0(x),\ x\in\IR$$ An integration of the above balance law with respect to t leads to a nonlinear Volterra integral equation of the form$$U(x,t)=\int\sbsp{0}{t} \lbrack H(x,\tau,U)-F\prime(U(x,\tau))U\sb{x}(x,\tau)\rbrack d\tau+U\sp0(x)$$ Sinc approximations to both derivative and the indefinite integral reduce the Volterra integral equation to an explicit system of algebraic equations of order 2N + 1. Thus, if the initial condition $U\sp0(x)$ is both analytic and of class L$\sb\alpha(D)$, then using various properties and facts of the Sinc function, it is shown that Sinc-Galerkin approximations produce an error of order $e\sp{-c/h}.$ Non-homogeneous Dirichlet boundary conditions are treated by subtracting the nonzero boundary conditions from the system using the change of variable $V(x,t)=U(x,t)-W(x,t),$ for some weight function $W(x,t).$ The method automatically handles the solution inside the elliptic region when the system changes type. Finally, many model problems over the domain $(-\infty,\infty)\times(0,T)$ are used to illustrate the accuracy and the implementation of the procedure.
Subject Area
Mathematics
Recommended Citation
Al-Khaled, Kamel Mustafa, "Theory and computation in hyperbolic model problems" (1996). ETD collection for University of Nebraska-Lincoln. AAI9637057.
https://digitalcommons.unl.edu/dissertations/AAI9637057