Abstract
The second‐order one‐dimensional linear hyperbolic equation with time and space variable coefficients and nonlocal boundary conditions is solved by using stable operator‐difference schemes. Two new second‐order difference schemes recently appeared in the literature are compared numerically with each other and with the rather old first‐order difference scheme all to solve abstract Cauchy problem for hyperbolic partial differential equations with time‐dependent unbounded operator coefficient. These schemes are shown to be absolutely stable, and the numerical results are presented to compare the accuracy and the execution times. It is naturally seen that the second‐order difference schemes are much more advantages than the first‐order ones. Although one of the second‐order difference scheme is less preferable than the other one according to CPU (central processing unit) time consideration, it has superiority when the accuracy weighs more importance.
Highlights
Second-order hyperbolic differential equations with variable coefficients are of common occurrence in mathematical physics, electromagnetic fluid dynamics, elasticity, and several other areas of science and engineering 1–7
Note that in 24, 25, the first- and second-order difference methods generated by an integer power of A are studied for solving the main equation in 1.1 for A t A with various nonlocal boundary conditions with respect to time variable
In 26, 27, the first- and second-order difference methods generated by an integer power of A are studied for solving the hyperbolic-parabolic equation for A t A with various nonlocal boundary conditions with respect to time variable
Summary
Second-order hyperbolic differential equations with variable coefficients are of common occurrence in mathematical physics, electromagnetic fluid dynamics, elasticity, and several other areas of science and engineering 1–7. A large cycle of research has been done on the finite difference schemes for the numerical solution of the special cases of the initial-value problem 1.1 ; see 8–11 and the references therein for example These methods are stable under the inequalities and contain the connection between the grid step sizes of time and space variables. Two different types of second-order difference methods considered in 22, 23 are introduced for solving the initial-value problem 1.1 Applying these difference schemes and the first-order difference scheme 1.2 , the numerical methods are supported in the third section by considering one-dimensional wave equation with time and space variable coefficients and nonlocal boundary conditions.
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