Differential equation series solution

The adsorbate concentration in the Nth stage along the charcoal bed can be found by solving the series of N differential equations. These solutions represent the concentration profile in the Nth stage. For a unit pulse of adsorbate at time t = 0, the solution reduces to... 

The formulation of heat conduction problems for the determination of the one-dimj .nsional transient temperature distribution in a plane wall, a cylinder, or a sphefeTesults in a partial differential equation whose solution typically involves irtfinite series and transcendental equations, wliicli are inconvenient to use. Bijt the analytical soluliop provides valuable insight to the physical problem, hnd thus it is important to go through the steps involved. Below we demonstrate the solution procedure for the case of plane wall. 

G. Adomian developed the decomposition method to solve the deterministic or stochastic differential equations. The solutions obtained are approximate and fast to converge, as shown by Cherrault In general, satisfactory results can be obtained by using the first few terms of the approximate, series solution. According to Adomian s theory, his polynomials can approximate the... 

Besides the intrinsic usefulness of Fourier series and Fourier transforms for chemists (e.g., in FTIR spectroscopy), we have developed these ideas to illustrate a point that is important in quantum chemistry. Much of quantum chemistry is involved with basis sets and expansions. This has nothing in particular to do with quantum mechanics. Any time one is dealing with linear differential equations like those that govern light (e.g. spectroscopy) or matter (e.g. molecules), the solution can be written as linear combinations of complete sets of solutions. 

Example Consider the differential equation for reaction and diffusion in a catalyst the reaction is second order c" — ac, c Qi) = 0, c(l) = 1. The solution is expanded in the following Taylor series in a. 

The series converge for all x. Much of the importance of Bessel s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. 

If the transverse loading is represented by the Fourier sine series in Equation (5.25), the solution to this fourth-order partial differential equation and subject to its associated boundary conditions is remarkably simple. As with isotropic plates, the solution can easily be verified to be... 

Dynamic methods may be classified as either classical, involving solution of Newton s equation, or quantal, involving solution of the (nuclear) Schrddinger equation (eq. (1.6)). Both of these are differential equations involving time, and can be solved by propagating an initial state through a series of small finite time steps. 

A simple repetition of the iteration procedure (2.20)-(2.22) results in divergence of higher order solutions. However, a perturbation theory series may be summed up so that all unbound diagrams are taken into account, just as is usually done for derivation of the Dyson equation [120]. As a result P satisfies the integral-differential equation... 

For convenience, mass fraction units are used for [I] and [M] instead of moles per unit volume to eliminate density, which is assumed constant. With an appropriate variable transformation and series expansion, the analytical solution of differential Equation 10 can be derived. The solution is as follows. 

Let us notice that due to orthogonality of Legendre s polynomials many functions can be represented as a series, which is similar to Equation (1.162), and this fact is widely used in mathematical physics. Now, we will derive the differential equation, one of the solutions of which are Legendre s functions. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

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... 

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... 

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

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... 

The only way to avoid this convergence problem is to terminate the infinite series (equation (G.49)) after a finite number of terms. If we let 2 take on the successive values / + 1, / + 2,..., then we obtain a series of acceptable solutions of the differential equation (G.43)... 

Attree RW, Cabell MJ, Cushing RL, Pieroni JJ (1962) A calorimetric determination of the half-life of thorium-230 and a consequent revision to its neutron capture cross section. Can J Phys 40 194-201 Bateman H (1910) Solution of a system of differential equations occurring in the theory of radioactive transformations. Proc Cambridge Phil Soc 15 423-427 Beattie PD (1993) The generation of uranium series disequilibria by partial melting of spinel peridotite ... 

The procedure for the solution of unsteady-state balances is to set up balances over a small increment of time, which will give a series of differential equations describing the process. For simple problems these equations can be solved analytically. For more complex problems computer methods would be used. 

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... 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

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