Linear Boundary-Value Problem

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

Pick s law of diffusion and Fourier s law of conduction are usually represented by second order ordinary differential equations (ODEs). In this chapter, we describe how one can obtain analytical solutions for linear boundary value problems using Maple and the matrix exponential. 

2 Exponential Matrix Method for Linear Boundary Value Problems 

Equation (3.1) can be converted into two first order differential equations (see section 2.1.4) and can be cast into matrix form as follows  

Start the Maple program with a restart command to clear all variables. 

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Due to its importance the impulse-pulse response function could be named. .contrast function". A similar function called Green s function is well known from the linear boundary value problems. The signal theory, applied for LLI-systems, gives a strong possibility for the comparison of different magnet field sensor systems and for solutions of inverse 2D- and 3D-eddy-current problems. 

Higher order terms require solving linear boundary value problems numerically. Here, as the estimate of P() effective for a large = p + -Tj2 f(y)dy + is used,... 

Apply the finite-difference method for solving a linear boundary value problem as follows Given the second derivative of the function y in the interval 0 to 4 as... 

Detailed instructions for a spreadsheet-based solution to this problem are found in Appendix D. This is a linear boundary-value problem that can be solved by any number of techniques, including analytical. However, the spreadsheet provides a relatively simple, fast, and efficient means to determine a solution. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

The approximation of Eq. (Bl) allows one to reduce Eqs. (A10) and (A11) to a linearized boundary-value problem (183,184,186). The latter can then be solved analytically and yields a compact matrix-form solution for the concentration profiles in the film region [58], Such a solution gives simple analytical expressions for the component fluxes with regard to the homogeneous reaction in the fluid films (see Ref. 135), which can be of particular value when large industrial reactive separation units are considered and designed. 

The boundary value problem (Eqs. (10), (11)) is usually solved numerically. However, it is also possible to use another approach employing a linearization of this second-order, non-linear problem and a subsequent analytical treatment The analytical solution of the linearized boundary value problem in the film region is obtained in [15] ... 

Linear Boundary Value Problems where the matrizant Q(A) is given by... 

The solution to equation (3.26) can be obtained by inverting the A matrix (X = A B). The procedure for solving linear boundary value problems using finite difference is as follows ... 

Solving Linear Boundary Value Problems Using Maple s dsolve Command... 

Maple s dsolve command can be used to solve linear boundary value problems. One of the advantages of using Maple s dsolve command is Maple can give Bessel and other special function solutions to linear boundary value problems. However, the analytical solution obtained from the dsolve command may not be in simplified or elegant form. The syntax for using the dsolve command is follows. 

In section 3.1.4, an analytical series solution using the matrizant was developed for the case where the coefficient matrix is a function of the independent variable. This methodology provides series solutions for Boundary value problems without resorting to any conventional series solution technique. In section 3.1.5, finite difference solutions were obtained for linear Boundary value problems as a function of parameters in the system. The solution obtained is equivalent to the analytical solution because the parameters are explicitly seen in the solution. One has to be careful when solving convective diffusion equations, since the central difference scheme for the first derivative produces numerical oscillations. 

In section 3.1.6, linear Boundary value problems were solved using Maple s dsolve command. The solution obtained may not be in the simplified form. Maple gives the Bessel function and other special function solutions for linear... 

Linear first order parabolic partial differential equations in finite domains are solved using the Laplace transform technique in this section. Parabolic PDEs are first order in the time variable and second order in the spatial variable. The method involves applying the Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes a boundary value problem (BVP) in the spatial direction with s, the Laplace variable as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 3.1 for solving linear boundary value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). Certain simple problems can be inverted to the time domain using Maple. This is best illustrated with the following examples. 

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