Periodicity Conditions for Third-Order Nonlinear With Application to Wave Propagation in Relaxing Media

It is known that wave propagation in nonlinear continues media, such as acoustic waves in solids, water waves, and solitary waves in arteries, can be reduced to a third order ordinary differential equations. They can be cast in a general third order ODE as x‴‴‴+f(t,x,x′,x″)=0. However, having an ODE as a reduced model for a phenomenon expressible by a partial differential equation lacks a proof to grantee for having a periodic solution. A third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the above general third-order ODE. However, the equation is too general. In this paper we examine the following more specific equation x‴‴‴+g1(x′)x″+g2(x)x′+g(x,x′,t)=e(t). and prove a new theorem to establish the sufficient condition for its periodicity. To obtain the periodicity conditions, the Schauder’s fixed-point theorem is implemented. A numerical method is also developed for rapid convergence.