In particular, we consider an initial-value problem for the nonlocal model with initial data strictly compatible with the solitary wave solution of the KdV equation or the BBM equation and then use a finite-difference scheme to solve the initial-value problem numerically.
In subsequent papers [7,8], global existence and nonexistence of solutions of the initial-value problems posed for various generalizations of the model were investigated.
We solve numerically the initial-value problems for (4) with the initial data
In the two numerical experiments, the exact travelling wave solutions to (6) and (9) are compared with the numerical solutions of the corresponding initial-value problems for (4).
This kind of initial-value problem
of ODEs can be solved using numerical computation.
Taylor's expansion of a second-order initial-value problem
The Taylor series expansion is an effective method to solve initial-value problems when the unknown functions have a Taylor expansion at an arbitrary point.
In fact, it can be used to solve initial-value problems involving nonlinear or linear ordinary differential equations of any order, or systems of such.
Equation (2) was first introduced in  and both global existence and blow-up results for solutions of the initial-value problem
with initial data in appropriate function spaces were established.
The first three chapters cover initial-value problems
, and boundary-value problems solved using discrete variable methods or finite element methods, for ordinary differential equations.
12] Mahmouda and Osman, MS: On a class of spline-collocation methods for solving second-order initial-value problems
1 Flow Topics Governed by Ordinary Differential Equations: Initial-Value Problems