Slope retention techniques for solving boundary-value problems in differential equations

Abstract

In the shooting technique for solving boundary-value problems for second-order ODE the initial (unknown) slope s = y′ ( a) at x = a is retained as a symbolic parameter during the first shot or pass. The parameter s is determined at the end of this pass by imposing the far boundary condition at x = b. One further pass using the now known value of s then determines the final solution. This technique can easily be generalized to higher-order ODE by retaining s = y′( a), t = y″( a), … as symbolic parameters and determining these from the boundary conditions at x = b. If the forward integration pass is performed with recurrence formulae such as the Numerov formula, then the parameters s, t, … are replaced by the parameters p = y( a+ h), q = y( a+2 h), … . Examples are given illustrating the application of the technique to second-order boundary-value problems, including non-linear problems and stiff problems. Eigenvalue problems are included within the framework of the present methods, with examples. Two examples are also given of the application of the technique to boundary-value problems in elliptic partial differential equations. The techniques employed illustrate the combined use of symbolic and numerical computation as a powerful tool in the solution of boundary-value problems.