Partial differential equations initial value type

The reason for constructing this rather complex model was that even though the mathematical equations may be easily set up using the dispersion model, the numerical solutions are quite involved and time consuming. Deans and Lapidus were actually concerned with the more complicated case of mass and heat dispersion with chemical reactions. For this case, the dispersion model yields a set of coupled nonlinear partial differential equations whose solution is quite formidable. The finite-stage model yields a set of differential-double-difference equations. These are ordinary differential equations, which are easier to solve than the partial differential equations of the dispersion model. The stirred-tank equations are of an initial-value type rather than the boundary-value type given by the dispersion model, and this fact also simplifies the numerical work. 

This set of hyperbolic partial differential equations for the gasifier dynamic model represents an open or split boundary-value problem. Starting with the initial conditions within the reactor, we can use some type of marching procedure to solve the equations directly and to move the solution forward in time based on the specified boundary conditions for the inlet gas and inlet solids streams. 

Equation (2.10) is an example of a parabolic second-order partial differential equation. The equation describes a single property, concentration, which evolves in space and time. In order to solve an equation of this type, we need to know the condition of the system at some starting time, f = 0. We have already stated that at the start of the experiment, the concentration of species A is a fixed value (1 mM for example) and is uniform everjrwhere. We call this the bulk concentration of species A and represent it with the symbol Ca. Therefore we have the initial condition ... 

We now have all the necessary tools to apply the Laplace transform method to linear partial differential equations, of the initial value type. The power of the Laplace transform in PDE applications is the ease with which it can cope with simultaneous equations. Few analytical methods have this facility. 

There are many numerical approaches one can use to approximate the solution to the initial and boundary value problem presented by a parabolic partial differential equation. However, our discussion will focus on three approaches an explicit finite difference method, an implicit finite difference method, and the so-called numerical method of lines. These approaches, as well as other numerical methods for aU types of partial differential equations, can be found in the literature [5,9,18,22,25,28-33]. 

The combination of these steps defines the temporal (<) and spatial (for typical onedimensional models x two- and three-dimensional simulations are also performed [10]) variation of the cmicentrations c of all educts, products, and intermediates involved. The concentrations at the electrode surface and in the bulk of the solution are defined through boundary conditions. Electrode boundary conditions also characterize the type of experiment performed, e.g., a triangular potential variation for cyclic voltammetiy or a potential or current step function for chronoampero- or chronopoten-tiometry, respectively. The equations of the resulting mathematical model form a system of partial differential equations (PDEs) with the c as the unknowns. This system is solved starting from initial values of the unknowns (initial conditions) and subject to the boundary conditions. The primary solution provides concentration profiles c =f(x) as a function of t. The experimental observable (in the case of a potential controlled experiment, the current i through the electrode ... 

Every partial differential equation needs an initial value or guess for numerical solver to start computing the equations. On the other hand, boundary conditions are specific for each conservation equation, described in Section 6.2. The variable in the continuity equation and momentum equations is the velocity vector, the variable in the energy equation is the temperature vector, and the variable in the species equation is the concentration vector. Therefore, appropriate velocity, temperature, and concentration values, which represent real-world values, need to be prescribed on each computational boundary, such as inlet, outlet, or wall at time zero. The prescribed values on boundaries are called boundary conditions. Each boundary condition needs to be prescribed on a node or line for 2D system or on a plane for 3D system. In general, there are several types of boundary conditions where the Dirichlet and Neumann boundary conditions are the most widely used in CFD and multiphysics applications. The Dirichlet boundary condition specifies the value on a specific boundary, such as velocity, temperature, or concentration. On the contrary, the Neumann boundary condition specifies the derivative on a specific boundary, such as heat flux or diffusion flux. Once the appropriate boundary conditions are prescribed to all boundaries on the 2D or 3D model, the set of the conservation equations is closed and the computational model can be executed. 

The partial differential equations defined in fhe previous two sections must be supplied boundary conditions. In general, there are two types of boundary condifions. Neumann conditions specify a flux entering the region and Dirichlet conditions place a constraint on a state variable at the boundary. In the examples in this chapter, we will specify the current density / at t > 0, fix fhe pofenfial af fhe anode side, and fix the water content at both anode and cathode sides. We will specify fhe initial conditions at time f = 0 that would exist if fhe current were zero. There, i/h will have a uniform value of The pofenfial everywhere is zero. We arbitrarily let X vary linearly across the membrane. These conditions are formulized in equation (8.32) ... 

Physical phenomena in science and engineering satisfy partial differential equations (PDEs), which relate changes in measurable quantities, like pressure or velocity, through partial derivatives taken in space and time. ETnlike ordinary differential equations (ODEs), whose integration constants are fixed by specifying values of the function and its derivatives at one or more points, PDEs require, in addition to functional information on curved boundaries, the specification of initial conditions for equations of evolution. Boundaries, we emphasize, may be external or internal, stationary or moving. The exact manner in which auxiliary conditions apply depends on the physical nature of the problem and is reflected in the type classification of the PDE studied. 

On the basis of their initial and boundary conditions, partial differential equations may be further classified into initial-value or boundary-value problems. In the first case, at least one of the independent variables has an open region. In the unsteady-state heat conduction problem, the time variable has the range 0 r >, where no condition has been specified at r = eo therefore, this is an initial-value problem. When the region is closed for all independent variables and conditions are specified at all boundaries, then the problem is of the boundary-value type. An example of this is the three-dimensional steady-state heat conduction problem described by the equation... 

Perhaps the simplest partial differential equation and set of boundary conditions is that of an equation which is first order in time and second order in a single spatial dimension, such as that of Eq. (12.4). The boundary conditions for such a problem are typically initial conditions in the time variable, i.e. the solution value at some initial time is known for the range of spatial variable and the solution then develops as a function of time and position. For the spatial variable, the boundary conditions are typically of the boundary value type where the solution value is known as a function of time on the boundaries of the spatial region. The spatial region of interest for such a problem may be either finite or of infinite extent. Equations that are second order in time such as Eq. (12.5) are also typically of the initial value type in time and of boundary value type in the spatial dimensions. 

With the above formulism a method is now defined for forming a finite difference set of equations for a partial differential equation of the initial value type in time and of the boundary value type in a spatial variable. The method can be applied to both linear and nonlinear partial differential equations. The result is an implicit equation which must be solved for the spatial variation of the solution... 

Many practical engineering problems involve not just a single variable and a single partial differential equation but a set of coupled partial differential equations. The extension of the finite differencing technique to coupled partial differential equations of the initial value, boundary value type is relatively straightforward and will be briefly discussed before developing code for solving such systems of equations. 

Listing 12.1. Code segment for single time step solution of a set of eoupled partial differential equations of the initial value, boundary value type. 

Some Examples of Partial Differential Equations of the Initial Value, Boundary Value Type... 

Seetion 12.5 has explored partial differential equations in two variables of the initial value, boundary value type. These typically arise in physical problems involving one spatial variable and one time variable. Several examples have been given of such practical problems. The present section is devoted to PDEs in two dimensions where boundary values are specified in both dimensions. Typically these two dimensions are spatial dimensions. Perhaps the prototype BVP is Poisson s equation which in two dimensions is ... 

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