Matrix exponential method

A series parallel reaction takes place in a CSTR (Bequette, 1998).[12] The governing equations are  

Find the time constant, x if the steady state concentration of A is 3 mol/liter (Hint Use the steady state version of the first equation to find the steady state concentration of A). Once x is obtained, plot the transient profile. 

This appendix presents two methods of obtaining an analytical solution to a system of first order ordinary differential equations. Both methods (power series and the Laplace transform) yield a solution in terms of the matrix exponential. That is, we seek a solution to 

Substitution of equation (A.2) into equation (A.l) with b = 0 yieids 

By equating coefficients of tike powers of t in equation (A.4) we obtain 

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Appendix A Matrix Exponential Method which yields... 

Solve the governing equations using the modified boundary conditions and obtain an analytical solution using the matrix exponential method. Show that one of the two boundary conditions at x = 1 becomes redundant (i.e., automatically satisfied). Plot the potential profiles inside the electrode for V = 5 = 1 and P = 0.1. 

Subtract the two equations in problem 6 to obtain a single equation for rj = Oi - O2. Obtain the boundary conditions for rj and arrive at an analytical solution for rj using the matrix exponential method. 

Obtain an analytical solution for this boundary value problem using matrix exponential method and plot the mole fraction profiles for ki = 1, k2 = 0.1. 

A more compact analytical solution can be obtained by using the matrix exponential method described in section 2.1.2. The dimensionless mass flux at X = 0 is given by ... 

In this chapter, we describe how one can arrive at the semianalytical solutions (solutions are analytical in the y variable and numerical in the spatial dimension) for linear elliptic partial differential equations using Maple and the matrix exponential method. 

Program Description The MATLAB function LinearODE.m solves a set of linear ordinary differential equations. The first part of the function checks the number of inputs and their sizes, or values. The next section of the function performs the solution of the set of ordinary differential equations, which can be done by either the matrix exponential method [Eq. (5.40)] or the eigenvector method [Eq. (5.53)]. The method of. solution may be introduced to the function through the fifth input argument. The default method of solution is the matrix exponential method. 

Use METHOD = 1 for matrix exponential method Use METHOD = 2 for eigenvector method Default value for METHOD is 1. 

Hence, as the second class of techniques, we discuss adaptive methods for accurate short-term integration (Sec. 4). For this class, it is the major requirement that the discretization allows for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This means, that we are interested in discretization schemes which avoid stepsize restrictions due to the fast oscillations in the quantum part. We can meet this requirement by applying techniques recently developed for evaluating matrix exponentials iteratively [12]. This approach yields an adaptive Verlet-based exponential integrator for QCMD. 

When considering the construction of exactly symmetric schemes, we are obstructed by the requirement to find exactly symmetric approximations to exp(—ir/f/(2fi,)). But it is known [10], that the usual stepsize control mechanism destroys the reversibility of the discrete solution. Since we are applying this mechanism, we now may use approximations to exp —iTH/ 2h)) which are not precisely symmetric, i.e., we are free to take advantage of the superior efficiency of iterative methods for evaluating the matrix exponential. In the following, we will compare three different approaches. 

Krylov Approximation of the Matrix Exponential The iterative approximation of the matrix exponential based on Krylov subspaces (via the Lanczos method) has been studied in different contexts [12, 19, 7]. After the iterative construction of the Krylov basis ui,..., Vn j the matrix exponential is approximated by using the representation A oi H(g) in this basis ... 

First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs). Single first order equations can be integrated by classical methods (Rice and Do, 1995). However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra. In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations. 

In this chapter anal5dical solutions were derived for linear ODEs using three methods the matrix exponential, Laplace transform, and dsolve. In section 2.1.2,... 

Appendix B Matrix Exponential by the Laplace Transform Method... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. In this chapter, we describe how one can arrive at the semianalytical solutions (solutions are analytical in the time variable and numerical in the spatial dimension) for linear parabolic partial differential equations using Maple, the method of lines and the matrix exponential. 

This set of coupled equations was solved by a modified exponential method (Chan et al., 1968). For an IV-dimensional basis set, a 21V-dimen-sional column matrix was defined with the first N elements given by the fi and the second N elements by dft/du. This matrix satisfies the differential equation... 

An alternative to split operator methods is to use iterative approaches. In these metiiods, one notes that the wavefiinction is fomially "tt(0) = exp(-i/7oi " ), and the action of the exponential operator is obtained by repetitive application of //on a function (i.e. on the computer, by repetitive applications of the sparse matrix... 

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

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