Equations, adjoint boundary value differential

B. If the reaction is second order and the numerical value of Da (Damkohler number) is the same as in part 1 (although its definition is slightly different), find the exit conversion using the Fox s iterative method (explain your formulation of adjoint equations for the iterative solution of the nonlinear two-point boundary-value differential equation). 

Important Note The iterative solution for this two-point boundary-value differential equation utilizing the adjoint equations is shown in Figure 4.13. Notice that this procedure may be unstable numerically and we may need to reverse the direction of the integration from the exit of the reactor toward the inlet as explained in the following subsection. 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

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