A computational procedure for solving a chemical flow reactor problem using shooting method (Q1805242)

The paper deals with the chemical flow reactor problem \(\varepsilon u'' + a(t) u' - b(t)u = f(t)\), \(0 < t < 1\), \(B_0 u(0) \equiv - u'(0) = \phi_1\), \(B_1 u(1) \equiv u(1) + \varepsilon u' (1) = \phi_2\), where \(\varepsilon > 0\) is a small parameter and \(a\), \(b\) and \(f\) are smooth functions satisfying \(a(t) \geq \alpha > 0\) and \(b(t) \geq 0\) with respect to \(t\). The problem is solved using the shooting method, i.e. the given problem is transformed into an initial value problem and then the resultant problem is solved numerically using a known exponentially fitted finite difference scheme. The similarity between the given chemical reactor flow problem and the initial value problem is that the nonuniformity occurs only in the derivative but not in the solution. The advantage of this method is that there is no need for matrix inversions as in tridiagonal difference schemes. The convergence of the method is correctly proved and some numerical results are given to illustrate the computational procedure.