Two Coupled PDEs

Consider the following highly coupled nonlinear PDEs[16] 

Equation (5.59) is chosen for illustration, because these equations are highly nonlinear, coupled, and also have an analytical solution  

Equation (5.59) is solved using the general procedure for nonlinear-coupled PDEs given below  

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

Call with(linalg) and with(plots) commands. 

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The drying of solids in general is a highly complex process involving both heat and mass transfer. If the solid is porous, moistme content and the temperatme will vary internally, as well as externally in the direction of airflow. Thus we could be dealing with at least two coupled PDEs (mass and energy balance) in time and two dimensions. 

Thus, we could be dealing with at least two coupled PDEs (mass and energy balance) in time and two dimensions. 

This approach is useful when dealing with relatively simple partial differential equation models. Seinfeld and Lapidus (1974) have provided a couple of numerical examples for the estimation of a single parameter by the steepest descent algorithm for systems described by one or two simultaneous PDEs with simple boundary conditions. 

Deterministic analysis Coupled biochemical systems Reaction kinetics are represented by sets of ordinary differential equations (ODEs). Rates of activation and deactivation of signaling components are dependent on activity of upstream signaling components. Spatially specified systems Reaction kinetics and movement of signaling components are represented by partial differential equations (PDEs). Useful for studies of reaction-diffusion dynamics of signaling components in two or three dimensions. (64-70)... 

We saw, in the previous section, that problems of creeping-motion in two dimensions can be reduced to the solution of the biharmonic equation, (7 46), subject to appropriate boundary conditions. To actually obtain a solution, it is convenient to express (7 46) as a coupled pair of second-order PDEs ... 

The first of these assumptions drops the momentum terms from the equations of motion, giving a situation known as creeping flow. This leaves and coupled through a pair of simultaneous PDEs. The pair can be solved when circumstances warrant, but the second assumption allows much greater simplification. It allows an uncoupling of the two equations so that is given by a single ODE ... 

Equation (8-62) generates three coupled linear second-order partial differential equations (PDEs). Eor complicated two-dimensional flow problems, this force balance and the equation of continuity yield three coupled linear PDEs for two nonzero velocity components and dynamic pressure. In some situations, this complexity is circumvented by taking the curl of the equation of motion ... 

If one approaches the solution of this problem via the equations of continuity and motion, then it is necessary to solve three coupled linear PDEs (i.e., one first-order PDE and two second-order PDEs) for Vx, Vy, and dynamic pressure. In the low-Reynolds-number limit, it is also possible to attack this problem via the three scalar components of the equation of change for finid vorticity. For example, if... 

At first glance, three coupled linear third-order PDEs must be solved, as illustrated above. However, each term in the x and y components of the vorticity equation is identically zero because =0 and Vj and Vy are not functions of z. Hence, detailed summation representation of the vorticity equation for creeping viscous flow of an incompressible Newtonian fluid reveals that there is a class of two-dimensional flow problems for which it is only necessary to solve one nontrivial component of this vector equation. If flow occurs in two coordinate directions and there is no dependence of these velocity components on the spatial coordinate in the third direction, then one must solve the nontrivial component of the vorticity equation in the third coordinate direction. 

In addition to the direct solution of PDEs corresponding to reaction-diffusion equations, in recent years attention has begun to be focused on the use of coupled lattice methods. In this approach, diffusion is not treated explicitly, but, rather, a lattice of elements in which the kinetic processes occur are coupled together in a variety of ways. The simulation of excitable media by cellular automata techniques has grown in popularity because they offer much greater computational efficiency for the two- and three-dimensional configurations required to study complex wave activity such as spirals and scroll waves. 

In Chapters 7 and 8, we presented numerical methods for solving ODEs of initial and boundary value type. The method of orthogonal collocation discussed in Chapter 8 can be also used to solve PDEs. For elliptic PDEs with two spatial domains, the orthogonal collocation is applied on both domains to yield a set of algebraic equations, and for parabolic PDEs the collocation method is applied on the spatial domain (domains if there are more than one) resulting in a set of coupled ODEs of initial value type. This set can then be handled by the methods provided in Chapter 7. 

We saw in the last example for the elliptic PDE that the orthogonal collocation was applied on two spatial domains (sometime called double collocation). Here, we wish to apply it to a parabolic PDE. The heat or mass balance equation used in Example 11.3 (Eq. 11.55) is used to demonstrate the technique. The difference between the treatment of parabolic and elliptic equations is significant. The collocation analysis of parabolic equations leads to coupled ODEs, in contrast to the algebraic result for the elliptic equations. 

The boundary condition implementations play a very critical role in the accuracy of the numerical simulations. The hydrodynamic boundary conditions for the LBM have been studied extensively. The conventional bounce-back rule is the most popular method used to treat the velocity boundary condition at the solid-fluid interface due to its easy implementation, where momentum from an incoming fluid particle is bounced back in the opposite direction as it hits the wall [20]. However, the conventional bounce-back rule has two main disadvantages. First, it requires the dimensionless relaxation time to be strictly within the range (0.5,2), otherwise the prediction will deviate from the correct result. Second, the nonslip boundary implemented by the conventional bounce-back rule is not located exactly on the boundary nodes, as mentioned before, which will lead to inconsistence when coupling with other partial differential equation (PDE) solvers on a same grid set [17]. 

In the preceding two sections, we discussed patterns in chemically reacting media using PDE reaction-diffusion models, where space, time, and chemical concentrations were continuous variables, and cellular automata, where space, time, and the state of a cell were discrete. We now turn our attention to coupled map lattices another type of model that has been used... 

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