Series solution

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  

Assume that y(x) can be expanded in a power series about x = 0  

We have redefined the summation index in order to retain the dependence on x . Analogously, 

Since this is true for all values of x, every quantity in square brackets must equal zero. This leads to the recursion relation 

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

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Linear elastic fracture mechanics (LEFM) is based on a mathematical description of the near crack tip stress field developed by Irwin [23]. Consider a crack in an infinite plate with crack length 2a and a remotely applied tensile stress acting perpendicular to the crack plane (mode I). Irwin expressed the near crack tip stress field as a series solution ... 

The decrease in with crack depth for fracture of IG-11 graphite presents an interesting dilemma. The utihty of fracture mechanics is that equivalent values of K should represent an equivalent crack tip mechanical state and a singular critical value of K should define the failure criterion. Recall Eq. 2 where K is defined as the first term of the series solution for the crack tip stress field, Oy, normal to the crack plane. It was noted that this solution must be modified at the crack tip and at the far field. The maximum value of a. should be limited to and that the far... 

The solution of the Laplace equation is not trivial even for relatively simple geometries and analytical solutions are usually not possible. Series solutions have been obtained for simple geometries assuming linear polarisation kinetics "" . More complex electrode kinetics and/or geometries have been dealt with by various numerical methods of solution such as finite differencefinite elementand boundary element. ... 

It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

One starts again with the Poincar6 series solution... 

We have entered into some details of the method of Poincar6 because it opened an entirely new approach to nonlinear problems encountered in applications. Moreover, the method is very general, since by taking more terms in the series solution (6-65), one can obtain approximations of higher order. However, the drawback of the method is its complexity, which resulted in efforts being directed toward a simplification of the calculating procedure. 

This shows that p (energy) increases exponentially at a fixed value of the phase Mathieu equation (6-127). Omitting the intermediate calculations (6-128) and (6-129), and taking instead of (6-130) the series solution in the form... 

There are two procedures available for solving this differential equation. The older procedure is the Frobenius or series solution method. The solution of equation (4.17) by this method is presented in Appendix G. In this chapter we use the more modem ladder operator procedure. Both methods give exactly the same results. 

The eigenvalues and eigenfimctions of the orbital angular momentum operator may also be obtained by solving the differential equation I ip = Xh ip using the Frobenius or series solution method. The application of this method is presented in Appendix G and, of course, gives the same results... 

Equation (6.24) may be solved by the Frobenius or series solution method as presented in Appendix G. However, in this chapter we employ the newer procedure using ladder operators. 

The Frobenius or series solution method for solving equation (G.l) assumes that the solution may be expressed as a power series in x... 

The interval of convergence for each of the series solutions u and ui may be determined by applying the ratio test. For convergence, the condition... 

We solve this differential equation by the series solution method. Applying equations (G.2), (G.3), and (G.4), we obtain... 

The ratio of consecutive terms in either series solution mi or U2 is given by the recursion formula with 5 = 0 as... 

A power series solution of equation (G.27) yields a recursion formula relating i+4, Ui+2, and ak, which is too complicated to be practical. Accordingly, we make the further definition... 

Agee, L. J., 1978, Power Series Solutions of the Thermal-Hydraulic Conservation Equations, in Transient Two-Phase Flew, Proc. 2nd Specialists Meeting, OECD Committee for the Safety of Nuclear Installations, Paris, Vol. 1, pp. 385-410. (3)... 

Figure 11-4 Flow control system. The components of the control system are linked in series. Solution...

The procedure described in Example 8-4 may be used to obtain analytical solutions for concentration profiles and tj for other shapes of particles, such as spherical and cylindrical shapes indicated in Figure 8.9. Spherical shape is explored in problem 8-13. The solution for a cylinder is more cumbersome, requiring a series solution in terms of certain Bessel junctions, details of which we omit here. The results for the dimensionless... 

In developing series solutions of differential equations and in other formal calculations it is often convenient to make use of properties of gamma and beta functions. The integral... 

On the other hand the series for y2(/t) does not terminate when n. > — 1 so there is no point in restricting n to be an integer. This series solution when multiplied by a factor... 

For two tanks in series, solution of the mass balance equations yields the result... 

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