DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

FINITE-DIFFERENCES METHOD FOR SOLVING OF
FIRST-ORDER HYPERBOLIC-TYPE EQUATIONS
WITH NONCONVEX STATE FUNCTION IN
A CLASS OF DISCONTINUOUS FUNCTIONS
MAHIR RASULOV AND BAHADDIN SINSOYSAL
A method for obtaining an exact and numerical solution of the Cauchy problem for a
first-order partial differential equation with nonconvex state function is suggested. For
this purpose, an auxiliary problem having some advantages over the main problem, but
equivalent to it, is introduced. On the basis of the auxiliary problem, the higher-order
numerical schemes with respect to time step can be written, such that the solution accurately
expresses all the physical properties of the main problem. Some results of the
comparison of the exact and numerical solutions have been illustrated.
Copyright © 2006 M. Rasulov and B. Sinsoysal. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Many important problems of physics and engineering are reduced to finding the solution
of equations of a first-order hyperbolic-type equations as
with the following initial condition
u t + F x (u) = 0 (1.1)
u(x,0)= u 0 (x). (1.2)
In this study, we consider Cauchy problem for a 1-dimensional first-order nonlinear
wave equation and propose a numerical method for obtaining the solution in a class of
discontinuous functions when F ′′ (u) has alternative signs.
2. The Cauchy problem for the nonconvex state function
As usual, let R 2 (x,t) be the Euclidean space of points (x,t). We denote Q T ={x ∈ R, 0≤
t ≤ T}⊆R 2 (x,t), here R = (−∞,∞).
Suppose that the function F(u) is known and satisfies the following conditions.
(i) F(u) is twice continuously differentiable and a bounded function for bounded u.
(ii) F ′ (u) ≥ 0foru ≥ 0.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 969–977