A fourth order cubic B-spline collocation method for the numerical study of the RLW and MRLW equations

This manuscript is concerned with a fourth order numerical method based on cubic B-spline functions to solve the regularized and modified regularized long wave(MRLW) equations which are very important nonlinear wave equations as they can be used to model a large number of problems arising in applied sciences. A rigorous Fourier series analysis has been carried out and it is shown that method is unconditionally stable. Efficiency and applicability of the method have been checked by applying it on seven important problems of the regularized long wave equation and modified regularized long wave equation. Such equations possess three invariants of motion viz. mass, energy and momentum which have been determined numerically and shown to coincide with their analytical values. The propagation of one, two and three solitary waves, undulations of waves and development of the Maxwellian initial condition into one, two and more solitary waves have been shown graphically. The main advantage of the method is that it captures propagation of solitary waves very efficiently. Method is capable of depicting the collisions between solitary waves. Conservation of mass energy and momentum is established by providing tables of related quantities. This method may be used to get solution at any point of the domain by using collocation formula. Since the numerical technique is unconditionally stable, time steps may be taken large so that solutions at higher time levels are calculated easily. Our mathematical results substantiate the physical relevance of the chosen model.