Differential equations power-series solution

The general power-series solution of the differential equation is, thus, given hy... 

So far we have considered only cases where the potential energy V(ac) is a constant. Hiis makes the SchrSdinger equation a second-order linear homogeneous differential equation with constant coefficients, which we know how to solve. However, we want to deal with cases in which V varies with x. A useful approach here is to try a power-series solution of the SchrSdinger equation. 

Section 4.1 Power-Series Solution of Differential Equations 63... 

We have seen in Chapter 3 that finite difference equations also arise in Power Series solutions of ODEs by the Method of Frobenius the recurrence relations obtained there are in fact finite-difference equations. In Chapters 7 and 8, we show how finite-difference equations also arise naturally in the numerical solutions of differential equations. 

This equation is characterized by regular singularities at the points x = 1 and at infinity. For m = 0, there are two linearly independent solutions to the Legendre differential equation and these solutions can be expressed as power series about the origin x = 0. In general, these series do not converge for... 

Not all differential equations of the general form (G.l) possess solutions which can be expressed as a power series (equation (G.2)). However, the differential equations encountered in quantum mechanics can be treated in this manner. Moreover, the power... 

In order to obtain well-behaved solutions for the differential equation (G.8), we need to terminate the infinite power series mi and U2 in (G.16) to a finite polynomial. If we let A equal an integer n (n = 0,1,2, 3,...), then we obtain well-behaved solutions... 

This result is the recursion formula which allows the coefficient an+2 to he calculated from the coefficient a . Starting with either ao or a an infinite series can be constructed which is even or odd respectively. These two coefficients are of course the two arbitrary constants in the general solution of a second-order differential equation. If one of them, say ao is set equal to zero, the remaining series will contain the constant at and be composed only of odd powers of On the other hand if a 0, the even series will result. It can be shown, however that neither of these infinite series can be accepted as... 

It is often possible to find a solution of homogeneous differential equations in the form of a power series. According to Frobenius, the power series should have the general form... 

The method of weighted residuals comprises several basic techniques, all of which have proved to be quite powerful and have been shown by Finlayson (1972, 1980) to be accurate numerical techniques frequently superior to finite difference schemes for the solution of complex differential equation systems. In the method of weighted residuals, the unknown exact solutions are expanded in a series of specified trial functions that are chosen to satisfy the boundary conditions, with unknown coefficients that are chosen to give the best solution to the differential equations ... 

We therefore advise that the reader should consult a recent series of papers published by Galvez et al. [171, 172] encompassing all the mechanisms mentioned in Sect. 7.1, elaborated for both d.c. and pulse polarography. The principles of the Galvez method are clearly outlined in the first part of the series [171]. It is similar to the dimensionless parameter method of Koutecky [161], which enables the series solutions for the auxiliary concentration functions cP and cQ exp (kt) and

combined directly with the partial differential equations of the type of eqn. (203). In some of the treatments, the sphericity of the DME is also accounted for. The results are usually visualized by means of predicted polarograms, some examples of which are reproduced in Fig. 38. Naturally, the numerical description of the surface concentrations at fixed potential are also immediately available, in terms of the postulated power series, and the recurrent relationships obtained for the coefficients of these series. 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

Adomian s Decomposition Method is used to solve the model equations that are in the form of nonlinear differential equation(s) with boundary conditions.2,3 Approximate analytical solutions of the models are obtained. The approximate solutions are in the forms of algebraic expressions of infinite power series. In terms of the nonlinearities of the models, the first three to seven terms of the series are generally sufficient to meet the accuracy required in engineering applications. 

To show that Equation 1.5 is a possible solution of Fick s second law, it can be substituted into Equation 1.4 and the differentiations performed (M and D are constant daX1 /3x = de /dx = atv3l leaxn, and duv/dx = udv/dx + vdu/dx). The solution of Equation 1.4 becomes progressively more difficult when more complex conditions or molecular interactions (which cause variations in D ) are considered. Indeed, many books have been written on the solutions to Equation 1.4, where Equation 1.5 actually represents thefirst term of a power series [note that (nD t)lfZ can be replaced by fnDji). 

Bessel s equation. Linear differential equation xly" + xy + x2 - tt2)y = 0, whose solutions are expressible as power series in x. 

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... 

In this appendix some important mathematical methods are briefly outlined. These include Laplace and Fourier transformations which are often used in the solution of ordinary and partial differential equations. Some basic operations with complex numbers and functions are also outlined. Power series, which are useful in making approximations, are summarized. Vector calculus, a subject which is important in electricity and magnetism, is dealt with in appendix B. The material given here is intended to provide only a brief introduction. The interested reader is referred to the monograph by Kreyszig [1] for further details. Extensive tables relevant to these topics are available in the handbook by Abramowitz and Stegun [2]. 

A very general method for obtaining solutions to second-order differential equations is to expand y(x) in a power series and then evaluate the coefficients term by term. We will illustrate the method with a trivial example that we have already solved, namely the equation with constant coefficients ... 

For regular singular points, a series solution of the differential equation can be obtained by the method of Frobenius. This is based on the following generalization of the power series expansion ... 

To start, we substitute the series expansion (Eq. 6.5) into the governing equation and the boundary condition (if the equation is a differential equation), and then apply a Taylor expansion to the equation and the boundary condition. Now, since the coefficients of each power of e are independent of e, a set of identities will be produced. This leads to a simpler set of equations, which may have analytical solutions. The solution to this set of simple subproblems is done sequentially, that is, the zero order solution y ix) is obtained first, then the next solution y,(x), and so on. If analytical solutions cannot be obtained easily for the first few leading coefficients (usually two), there is no value in using the perturbation methods. In such circumstances, a numerical solution may be sought. 

We note that 0i and 02 are the phases of the extraordinary and ordinary waves, respectively. With 0i and 02 given by Eqs. (12a) and (12b), Eq. (11) reduces to a set of two second-order differential equations for the coeffleients Ci and C2 only. The solution of this set of equations can be found by first expanding C] and C2 into power series of 1/koi... 

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