Differential equation, linear, boundary particular solutions

To meet a particular application, known solutions of a linear differential equation may be combined to meet the boundary conditions of that application. The superposition principle will be demonstrated through its use in Example 2.4. 

Because our interest is with second-order differential equations, two linearly independent solutions always arise (the Wronskian of solutions is non-zero [490], see Sect. 5) and requires two arbitrary constants to be fixed from the two boundary conditions imposed on p0(r, t) by the physics of the problem being modelled. These boundary conditions determine how much of each of the two linearly independent solutions of the homogeneous equation (317) must be added to the particular integral to ensure that the solution of eqn. (316) is consistent with the boundary conditions. In the next three sections, the method of deriving the particular integral from the two linearly independent solutions of the homogeneous equation are discussed. 

The set of coupled equations together with the conditions for R — 0 and R — oo constitute a boundary value problem. It does not suffice merely to find a solution of the coupled equations the solution must also have a particular behavior as R goes to zero and to infinity. One possible way of solving the boundary value problem is the transformation into a computationally more convenient initial value problem (Lester 1976). From the theory of linear differential equations we know that Equation (3.5) has N regular solutions, i.e., solutions which diminish as R — 0, and which are linearly independent. Each solution has a distinct behavior in the limit R —> oo which, however, is not necessarily the behavior imposed by (3.50). However, if we have found one particular set of solutions, we can easily construct new solutions by taking linear combinations so that the new wavefunctions fulfil the required boundary conditions. 

These boundary conditions are particularly convenient to evaluate the integration constants, as illustrated below. The mass transfer equation corresponds to a second-order linear ordinary differential equation with constant coefficients. The analytical solution for I a is... 

Equations 8.5.1.3 are quite general they involve no assumptions regarding the constancy of particular matrices they apply to mixtures with any number of components and for any relationship between the fluxes. It is at this point where any assumptions necessary to solve Eqs. (8.5.1-8.5.3) must be made. In the three methods to be discussed below we proceed in exactly the same way as we did when deriving the exact solution and the solution to the linearized equations first obtain the composition profiles, then differentiate to obtain the gradients at the film boundary, and combine the result with Eq. 8.5.3 to obtain the working flux equations. 

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