Differential equations, solution adaptive

D.F. Hawken, J.J. Gottlieb, and J.S. Hansen, Review of Some Adaptive Node-Movement Techniques in Finite-Element and Finite-Difference Solutions of Partial Differential Equations, J. Comput. Phys. 95 (1991). 

A generalized partial differential equation solver which handles simultaneous parabolic, one dimensional elliptic, ordinary and integral equations and uses B-splines with an adaptive grid was written to solve the model. Further details on the model and solution method can be found in Reference 14. 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

Mathematically, studies of diffusion often require solving a diffusion equation, which is a partial differential equation. The book of Crank (1975), The Mathematics of Diffusion, provides solutions to various diffusion problems. The book of Carslaw and Jaeger (1959), Conduction of Heat in Solids, provides solutions to various heat conduction problems. Because the heat conduction equation and the diffusion equation are mathematically identical, solutions to heat conduction problems can be adapted for diffusion problems. For even more complicated problems, including many geological problems, numerical solution using a computer is the only or best approach. The solutions are important and some will be discussed in detail, but the emphasis will be placed on the concepts, on how to transform a geological problem into a mathematical problem, how to study diffusion by experiments, and how to interpret experimental data. 

Our solution technique will be to adapt the governing equation and the boundary conditions from Example 2.2 to apply to this problem. We spent considerable effort proving to ourselves that equation (E2.2.3) was a solution to the governing differential equation. We can avoid that tedious process by showing that this problem has a similar solution, except with different variables and different constants. 

The system of hyperbolic and parabolic partial differential equations representing the ID or 2D model of monolith channel is solved by the finite differences method with adaptive time-step control. An effective numerical solution is based on (i) discretization of continuous coordinates z, r and t, (ii) application of difference approximations of the derivatives, (iii) decomposition of the set of equations for Ts, T, c and cs, (iv) quasi-linearization of... 

This is Fick s second diffusion equation [242], an adaptation to diffusion of the heat transfer equation of Fourier [253]. Technically, it is a second-order parabolic partial differential equation (pde). In fact, it will mostly be only the skeleton of the actual equation one needs to solve there will usually be such complications as convection (solution moving) and chemical reactions taking place in the solution, which will cause concentration changes in addition to diffusion itself. Numerical solution may then be the only way we can get numbers from such equations - hence digital simulation. 

We also use a restricted form of Equation 19 for the kinetics studies described previously. Smog chamber analyses uses just the first and last terms so that they depend on ordinary differential equations. These are solutions which describe the time-dependent behavior of a homogeneous gas mixture. We used standard Runge-Kutta techniques to solve them at the outset of the work, but as will be shown here, adaptations of Fade approximants have been used to improve computational efficiency. 

In Eq. (5.8.12) we already specified H in terms of T, Pm, o- The determination of G in the same variables is more involved we base our derivation on Eq. (1.13.19) adapted to the present situation. This is actually an ordinary differential equation of standard form since all variables save T are fixed. Invoking Eq. (1.3.27) as the solution to the first order differential equation (1.13.19) one obtains the expression... 

Dissinger, G. R. GRD1 - A New Implicit Integration Code for the Numerical Solution of Partial Differential Equations on Either Fixed or Adaptive Spatial Grids, Doctoral Dissertation, Lehigh Univ. Bethlehem, PA, 1983. 

C. Johnson. Numerical solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1990. An excellent introductory book on the finite element method. The text assumes mathematical background of first year graduate students in applied mathematics and computer science. An excellent introduction to the theory of adaptive methods. 

Three solution algorithms based on the method of lines for systems of parabolic differential equations cire tested by simulation of a reversed flow reactor for exhaust air purification. The solutions are compared with regcird to solution quality and computing time needed. It will be shown, that only the fully adaptive method will guarantee a sufficient solution quality. 

The comparison of three different solution methods with and without spatial and temporal adaptivity for systems of parabolic differential equations has shown the necessity of using a fully adaptive solution method to achieve authentic solutions. The fully adaptive method liberates the user from choosing grids and time stepsizes. This saves much time because only one solution has to be computed, no comparison of different solutions is necessary. By choosing timesteps and grids which promise optimal performance, the computation time needed for adaptive methods can be optimized. Comparing the solutions of identical quality, the fully adaptive method clearly outperforms its alternatives for the examples considered. 

Petzold, L. R., An adaptive Moving Grid Method for One-Dimensional Systems of Partial Differential Equations and its Numerical Solution, Proc. Workshop on Adaptive Methods for Partial Differential Equations, Renselaer Polytechnic Institute (1988)... 

The numerical solution is performed by the method of lines. Spatial discretization of the partial differential-equation system using finite differences on statically adapted grids leads to large systems of ordinary differential and algebraic equations. This system of coupled equations is solved by an implicit extrapolation method using the software package LIMEX [14]. The code computes species mass-fraction and temperature profiles in the gas phase, fluxes at the gas-surface interface, and surface temperature and coverage as function of time. 

Thompson JF (1985) A survey of dynamically-adaptive grids in the numerical solution of partial differential equations. Appl Numer Math 1 3-27... 

The theory of deterministic linear systems plays a fundamental role in the dynamic analysis of structures subjected to stochastic excitations. For this reason in this section the fundamental of deterministic analysis of SDoF subjected to deterministic excitation is synthetically reviewed. Particular care has been devoted to the state-space approach. This approach is the best suited for the development of formulations in the framework of random vibrations. In fact, its adaptability to numerical method of solution of differential equations and its extension to multi-degree-of-freedom (MDoF) systems are very straightforward. 

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