Exponential matrix method, linear

Maple s dsolve command was used to solve linear ODEs in section 2.1.6. In our opinion, exponential matrix method is the best method to arrive at an elegant analytical solution. The Laplace transform technique illustrated in section 2.1.5 could be used for integro-differential equations. Maple s dsolve command has to be used if the exponential matrix method fails. 

Exponential Matrix Method for Linear Boundary Value Problems... 

Steady state heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear elliptic partial differential equation. For linear parabolic partial differential equations, finite differences can be used to convert to any given partial differential equation to system of linear first order ordinary differential equations in time. In chapter 5.1, we showed how an exponential matrix method [3] [4] [5] could be used to integrate these simultaneous equations... 

If the displacements are small the quadratic terms can be dropped. Using the matrix method of Chapter 5 and assuming that f and rj are linear combinations of decaying exponential functions e the decay constants X are solutions of the determinantal equation... 

We use an alternative to this method, that enables a fast and accurate evaluation of the two-center integrals. Analytical integration is possible when linear combinations of Gaussian Type Orbitals are used to describe the atomic states [1,9, 10]. Imperfect behavior of such gaussian functions at large distances does not affect the results, since the two-center matrix-elements (7) have an exponential decay for increasing intemuclear distances. For example, for integrals and expressed in cartesian coordinates, one has to evaluate expressions such as... 

In this section, we have applied our novel method to the detecting ordered motions with the linear increase of singular values of stability matrixes. From the crossover times of local instabilities that change from linear to exponential increases, lifetimes of... 

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,... 

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. 

Our method makes it possible to use simple linear rules for exploring complicated nonlinear systems. A simple application is the study of connectivity among various chemical species in complicated reaction networks. In the simple case of homogeneous systems with time-invariant structure, the susceptibility matrix x = [Xhh ] = X depends only on the transit time and not on time itself. The matrix elements Xuu ( ) are proportional to the elements (t) of a Green function matrix G (t) = [G / (t)], which is the exponential of a connectivity matrix K, that is, G (t) = exp [tK]. It follows that from a response experiment involving a system with time-invariant structure, it is possible to evaluate the connectivity matrix, K, which contains information about the relations among the different chemical species involved in the reaction mechanism. The nondiagonal elements of the matrix K = Kuu I show whether in the reaction mechanism there is a direct connection between two species in particular, if Kuu 0, there is a connection from the species u to the species u the reverse connection, from u io u, exists if Ku u 0. 

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. 

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