PDEs functional domains

PDEs are generally differentiated on the basis of their substrate specificity and how they are regulated. They consist of three main functional domains a regulatory C-terminus (probably involved in the actions of PDE-specific kinases), a central catalytic domain, and a regulatory N-terminus (involved in the allosteric regulation of substrate binding and phosphorylation and membrane targeting). 

The finite difference method (FDM) is probably the easiest and oldest method to solve partial differential equations. For many simple applications it requires minimum theory, it is simple and it is fast. When a higher accuracy is desired, however, it requires more sophisticated methods, some of which will be presented in this chapter. The first step to be taken for a finite difference procedure is to replace the continuous domain by a finite difference mesh or grid. For example, if we want to solve partial differential equations (PDE) for two functions 4> x) and w(x, y) in a ID and 2D domain, respectively, we must generate a grid on the domain and replace the functions by functions evaluated at the discrete locations, iAx and jAy, (iAx) and u(iAx,jAy), or 4>i and u%3. Figure 8.1 illustrates typical ID and 2D FDM grids. 

They also found that only needs to be a distance function close to the front. Therefore, it is not necessary to solve the PDE to steady state over the whole domain. We may then locate the interface as the local areas within the calculation domain where ]d] < e, and we iterate on d until ]V i = 1 near the interface. The front will then have a uniform thickness, as this solution corresponds to [V ] = 1 when

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It remains to determine the four sets of constants, An, Bn, Cn, and Dn, and the function fo that describes the 0(Ca) correction to the shape of the drop. For this, we still have the five independent boundary conditions, (7-207)-(7-210). It can be shown that the conditions (7-207), (7-208), and (7-209) are sufficient to completely determine the unknown coefficients in (7-213) and (7-215). Indeed, for any given (or prescribed) drop shape, the four conditions of no-slip, tangential-stress continuity and the kinematic condition are sufficient along with the far-field condition to completely determine the velocity and pressure fields in the two fluids. The normal-stress condition, (7-210), can then be used to determine the leading-order shape function /0. Specifically, we can use the now known solutions for the leading-order approximations for the velocity components and the pressure to evaluate the left-hand side of (7 210), which then becomes a second-order PDE for the function /(). The important point to note is that we can determine the 0(Ca) contribution to the unknown shape knowing only the 0(1) contributions to the velocities and the pressures. This illustrates a universal feature of the domain perturbation technique for this class of problems. If we solve for the 0(Cam) contributions to the velocity and pressure, we can... 

Fig. 17.3. Schematic representation of the domain organization of PDEs. The conserved catalytic domain is found toward the C-terminal half of the PDE (8). The number and function of the reguiatory domains differ between the different PDE subfamilies. These domains include calciumsensing domains as well as signaling domains that mediate interactions with other signaling proteins.

The projection-based model order reduction algorithm begins with a spatial discretization of the governing PDEs to attain the dynamic system equations as Eq. 11. Specifically, here, X(t) is the state vector of unknowns (a function of time) on the discrete nodes, n is the total number of nodes A is formulated by the numerical discretization Z defines the functions of boundary conditions and source terms and B relates the input function to each state X. Equation 11 can be recast into the frequency domain in terms of transfer function T(s). T(s) then is expanded as a Taylor series at s = 0 yielding... 

After assuming a flow model based both on the physical structure of the reactor and the characteristic features of the experimental RTD curve, mass balance equations are written for the dispersion of an inert tracer among the various zones involved in the model. These equations are linear differential equations (ODE or PDE) owing to the linear character of mixing processes. By solving the equations in the Laplace domain, the theoretical transfer function G (s,p ) is obtained, which is nothing but the Laplace transform of the - theoretical RTD E (t,pj ) where pj are the parameters of the model... 

The z-dependent transfer function C2(z, s) depends on the boundary condition of flic PDE Eq. 21. Using Eq. 23 and the field Eqs. 16 and 17, one can derive the expressions for the electric field E and then for the electric potential in the L lace domain ... 

Alternatively, in a real-space method, we specify a munber of grid points rbl e and compute numerically the field values (p r, t) at these points. While function-space methods can yield analytical solutions for some linear PDEs in simple domains, real-space methods require numerical solution yet are more generally apphcable, especially for problems with nonlinear source terms or complex domain geometries. We thus restrict our focus to real-space methods, although the finite element method will be seen to mix both approaches. For further discussion of function-space approaches, consult Bird et al. (2002), Deen (1998), and Stakgold (1979). 

The optional MATLAB PDF toolkit (doc pdetool), created by tiie developers of FEMLAB (www.comsol.com), has tools for forming meshes and solving simple PDEs in two dimensions, pdetool opens a graphical nser interface (GUI), in which we can draw the domain, mesh it, specify boundary conditions and PDE parameters, solve, and plot the solution. As tutorials are provided on die use of the GUI, here our focus is upon use of the command-line interface to access the functions of the PDE toolkit directly. 

As wilt be subsequently developed, the shape functions and local coordinates play very important roles in formulating a set of matrix equations for a PDE over a spatial domain wifli flie FE method while constant coordinate lines for f 2 and f 3 would be parallel to the other two sides of the triangle. 

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