Partial Differential Equations and Special Functions

Chapter 12 Partial Differential Equations and Special Functions... 

In order to solve a partial differential equation using the PDE toolbox, one may simply use the graphical user interface by employing the pdetool command. In this separate environment, the user is able to define the two-dimensional geometry, introduce the boundary conditions, solve the partial differential equation, and visualize the results. In special cases where the problem is complicated or nonstandard, the user may wish to solve it using command-line functions. Some of these functions (solvers only) are listed in Table 6.4. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

Equation 3.42 is a partial differential equation that will, upon solution, yield concentration as a function of time and distance for any sample pulse undergoing uniform translation and diffusion. In theory, we need only specify the initial conditions (i.e., the mathematical shape of the starting peak) along with any applicable boundary conditions and apply standard methods for solving partial differential equations to obtain our solutions. These solutions tend to be unwieldy if the initial peak shape is complicated. Fortunately, a majority of practical cases are described by a relatively simple special case, which we now describe. 

In section 4.4, the given linear parabolic partial differential equation in semi-infinite domain was solved by combining the independent variables (similarity solution). This technique is capable of providing special function solutions as shown in example 4.9. In section 4.5, elliptic partial differential equations were solved using the similarity solution technique. In section 4.6, similarity solution was extended for nonlinear parabolic and elliptic partial differential equations. 

In section 7.1.7, eigenfunctions and eigenvalues were obtained numerically. This method is very general and can be used to avoid the use of complicated special function solutions. In section 7.1.8, the separation of variables method which was illustrated earlier for parabolic partial differential equations was extended to elliptic partial differential equations. A total of fourteen examples were presented in this chapter. 

Because of the complicated nature of biomolecular geometries and charge distributions, the PB equation (PBE) is usually solved numerically by a variety of computational methods. These methods typically discretize the (exact) continuous solution to the PBE via a finite-dimensional set of basis functions. In the case of the linearized PBE, the resulting discretized equations transform the partial differential equation into a linear matrix-vector form that can be solved directly. However, the nonlinear equations obtained from the full PBE require more specialized techniques, such as Newton methods, to determine the solution to the discretized algebraic equation. ... 

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