Solution of third order viscous wave equation using finite difference method

Abstract

Abstract The aim of this paper was to solve the third order viscous wave equation: utt = vuxx + c uxxt, which is a PDE. It occurs in many real-life situations such as water waves, sound waves, radio waves, light waves and seismic waves. This equation has been solved before using analytical methods but not yet been exhaustively nor conclusively done. Two schemes, namely CD-FD and CN-FD were developed and the equation discretised by FDM. We used each scheme respectively to obtain solution algorithms. Stability of the schemes was analysed, consistency of the numerical solutions with the original equation was tested, and Mathematica software used to generate solutions. The numerical computational results obtained for solutions of third order viscous wave equation obtained for varying the mesh ratio showed that the schemes were both conditionally stable and consistency noted. We found that as the mesh ratio reduces, the solution tends towards the exact solution. The solution algorithm showed consistency with the original viscous equation when tested. In addition, the equation simulates many physical situations which include designing of bridges, acoustics, gas dynamics, seismology, meteorology among many other natural phenomena. This work contributes to mathematical knowledge in research and innovations which apply PDEs.