The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude, long waves in dispersive media. This evolution equation has been used in both laboratory and field studies of water waves. The principal new result is an exact theory of convergence of the two-point boundary-value problem to the initial-value problem posed on an infinite stretch of the medium of propagation. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary-value problem in numerical studies of the initial-value problem posed on the entire line.

In this paper, the nonlinear system of initial value problems are solved numerically by using Residual method which is based on the minimizing residual function by the Taylor's series expansion. The convergence analysis of the method is given. The signi, cant feature of the method is reduction of nonlinear system of initial value problems to the system of linear equations. To emphasize the accuracy and poten-tial of the method, we solve Lorenz system and primary HIV-1 infection problem numerically.

In this paper an improved version of the collocation method is proposed to solve ordinary di, erential equations with initial conditions. Our proposed algorithm is described by applying it to some well-known IVPs. The results are compared with basic collocation algorithms to show the advantages, applicability and e, ciency of the proposed method. Based on numerical results, the proposed algorithm has better accuracy and execution time.

We are concerned with the development of a k-step method for the numerical solution of fuzzy initial value problems. Convergence and stability of the method are also proved in detail. Moreover, a specific method of order 4 is found. The numerical results show that the proposed fourth order method is efficient for solving fuzzy differential equations.