Boundary value problems diffusion equation

The numerical method of lines described in the previous example can be used for nonlinear elliptic partial differential equations, also. For example, consider the following nonlinear boundary value problem (diffusion with a second order reaction) ... 

This treatment of reaction at the limit of bulk diffusion control is essentially the same as that presented by HugoC 69j. It is attractive computationally, since only a single two-point boundary value problem must be solved, namely that posed by equations (11.15) and conditions (11.16). This must be re-solved each time the size of the pellet is changed, since the pellet radius a appears in the boundary conditions. However, the initial value problem for equations (11.12) need be solved only once as a preliminary to solving (11.15) and (11.16) for any number of different pellet sizes. 

In practice it would not be reasonable to solve the balances at the limit of Knudsen diffusion control by considering the n simultaneous boundary value problems (11.7). All the partial pressures can be expressed in terms of by integrating equations (11,25), with the result... 

Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffusivity is constant, the solution is the error function, a tabulated function. 

For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a boundary value problem . The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism ... 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

Solution. Yes. When D varies with concentration we have shown in Section 4.2.2 that the diffusion equation can be scaled (transformed) from zt-space to 77-space by using the variable rj = x/ /4Di (see Eq. 4.19). Also, under diffusion-limited conditions where fixed boundary conditions apply at the interfaces, the boundary conditions can also be transformed to 77-space, as we have also seen. Therefore, when D varies with concentration, the entire layer-growth boundary-value problem can be transformed into 77-space. Since the fixed boundary conditions at the interfaces require constant values of 77 at the interfaces, they will move parabolically. 

C. Boundary Value Problems for the Fractional Diffusion Equation HI. The Fractional Fokker-Planck Equation... 

Exemplifying the convenience of the fractional approach, we address the imposition of boundary value problems on the fractional diffusion equation which was demonstrated in Ref. 62. In this force-fiee case for which the kernel, Eq. (27), takes on the homogeneous form K(x,x u) = uw(u) (2( c — x ) — <5(x))/(l - w(u)), one can apply the method of images in order to construct the solution [12]. 

The differential equations and the boundary value problem (bvp) which can be fulfilled by rR and c 1 if the thickness of the membrane is greater than the diffusion layer of X+ into the membrane are ... 

The differential diffusion equations system to solve when a potential pulse E is applied and the corresponding boundary value problem (bvp) when the expanding plane model for the DME is considered are ... 

Under these conditions (see Eqs. (4.199)-(4.202)), it can be easily demonstrated that the Superposition Principle can be applied and the diffusion differential equation and the boundary value problem of this process, independently of the electrode geometry, are simplified to... 

The composition boundary values entering into Eqs. (All) represent external values for Eqs. (A10). With some further assumptions concerning the diffusion and reaction terms, this allows an analytical solution of the boundary-value problem [Eqs. (A10) and (All)] in a closed matrix form (see Refs. 58 and 135). On the other hand, the boundary values need to be determined from the total system of equations describing the process. The bulk values in both phases are found from the balance relations, Eqs. (Al) and (A2). The interfacial liquid-phase concentrations xj are related to the relevant concentrations of the second fluid phase, y , by the thermodynamic equilibrium relationships and by the continuity condition for the molar fluxes at the interface (57,135). 

The pore scale boundary value problem is described by the advection-diffusion equation... 

The Mean Transport Pore Model (MTPM) described diffusion and permeation the model (represented as a boundary value problem for a set of ordinary differential equations) are based on Maxwell-Stefan diffusion equation and Weber permeation law. Parameters of MTPM are material constants of the porous solid and, thus, do not dependent on conditions under which the transport proeesses take place. 

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. 

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. 

Boundary value problems in cylindrical and spherical coordinates have an inherent singularity at x = 0. These problems can be tackled using Maple s inbuilt midpoint methods. For example, diffusion of a substrate in an enzyme catalyzed reaction.[6] The governing equation for the dimensionless concentration is... 

In the early 1970s, the standard finite element approximations were based upon the Galerkin formulation of the method of weighted residuals. This technique did emerge as a powerful numerical procedure for solving elliptic boundary value problems [102, 75, 53, 84, 50, 89, 17, 35]. The Galerkin finite element methods are preferable for solving Laplace-, Poisson- and and diffusion equations because they do not require that a variational principle exists for the problem to be analyzed. However, the power of the method is still best utilized in systems for which a variational principle exists, and it... 

The general solution to the differential equation includes many possibilities the engineer needs to provide initial conditions to specify which solution is desired. If aU conditions are available at one point [as in Eq. (8.1)], then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equation becomes a two-point boundary value problem (see Chapter 9). Initial value problems as ordinary differential equations arise in the control of lumped parameter models, transient models of stirred tank reactors, and generally in models where there is no diffusion of the unknowns. 

The analytical solutions presented above are most of all derived on the basis of the very simple Henry isotherm or the more physically sensible Langmuir isotherm. Beside these analytical solutions a direct integration of the initial and boundary value problem of the diffusion-controlled model is possible. To do so differentials are replaced by differences. This approximation leads to linear equation systems for each time step which have to be solved. As... 

Eq. (5.248) and its modification for a deformed surface [154], together with the corresponding equations for AT [152] and Eq. (5.243) are the only analytical results obtained as solution of the boundary value problem for the diffusion equations of micelles and monomers. An approximate relation for Ay can be also obtained without integration of the diffusion equations with the help of the penetration theory [155], In this case the derivative on the right hand side of Eq. (5.237) is replaced by the ratio of finite differences... 

Application of numerical methods have been rather seldom in studies of adsorption kinetics from micellar solutions. The main difficulties are probably connected with the large number of independent parameters. The first work belongs to Miller [146]. Fainerman and Rakita also published numerical results of the solution of the boundary value problem (5.236), (5.237), (5.245) [85]. Recently Danov et al. proposed an original method for solving the boundary value problem for the diffusion of micelles and monomers [92]. The system of equations was reduced to a system of ordinary differential equations by using a model concentration profile in the bulk phase. The obtained results agree better with dynamic surface tensions of micellar solutions than equation (5.248). 

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