Solution by Separation of Variables

It is often possible to write the solution of a partial differential equation as a sum of terms, each of which is a function in one of the variables only. This procedure is called solution by separation of variables. The one-dimensional wave equation... 

D. Start-Up Flow in a Circular Tube - Solution by Separation of Variables... 

D. START-UP FLOW IN A CIRCULAR TUBE - SOLUTION BY SEPARATION OF VARIABLES... 

Now knowing the solution for the overall mass balance, we seek a solution for the component balance. It can also be rearranged for solution by separation of variables. Expanding the left-hand side of equation (1.3.8) gives... 

An exact solution by separation of variables has been provided by Soo and Rodgers (1971). The solution for small values of and a (Soo, 1989) is... 

Solution of this linear differential equation by separation of variables and subsequent mathematical manipulation leads to... 

Analytical solution is possible only when the reaction in the body of the reactor is first or zero order, otherwise a numerical solution will be required by finite differences, method of lines or finite elements. The analytical solution proceeds by separation of variables whereby the PDE is converted into ODEs whose solutions are in terms of trigonometric functions. Satisfying all of the boundary condtions makes the solution of the PDE an infinite series whose development is quite elaborate and should be sought in books on Fourier series or partial differential equations. 

For a given olivine crystal with radius a, treat it as a plane sheet along the c-axis with half-thickness a. Assume that the initial zonation is symmetric with respect to the center. Approximate the initial profile as C = Cq + CiCOs(7ix/a), where Q is the amplitude of the variation. Assume no flux boundary condition. The solution to the diffusion problem can be found by separation of variables as... 

It suffices to show that K Pl (V- -) = 0. So suppose that f e K fi (V" "), i.e., suppose that f and its first and second partial derivatives are continuous, that Df = 0 and that f is orthogonal to every solution obtained by separation of variables. We will show that f = 0. 

The parabolic partial differential equation can be solved by separation of variables, although the solution shown in Fig. 4.9 is found by a finite-difference technique. Starting from rest (i.e., zero velocity everywhere), the expected steady-state parabolic velocity profile is reached in a dimensionless time of t 1. 

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

This equation can be solved by separation of variables, provided the potential is either a constant or a pure radial function, which requires that the Lapla-cian operator be specified in spherical polar coordinates. This transformation and solution of Laplace s equation, V2 / = 0, are well-known mathematical procedures, closely followed in solution of the wave equation. The details will not be repeated here, but serious students of quantum theory should familiarize themselves with the procedures [15]. 

This formidable-looking equation can be solved by separation of variables. Assume that the solution is a product of two independent functions of the three variables ... 

The solution to this ordinary differential equation may be written down by separation of variables as... 

In fact, an exact solution of the Graetz problem can be achieved by separation of variables.17 It is convenient to pose the problem in terms of a new unknown temperature function... 

A general solution of this equation, which is bounded at ij = 1, can be obtained easily by separation of variables in terms of modified Bessel functions of the first and second kind... 

First, we consider a parallelepiped with sides L, L2, and L3. The solution of the corresponding three-dimensional problem (4.2.1)-(4.2.3) can be constructed by separation of variables and results in the following formula for bulk temperature [277] ... 

For long diffusion timas, where the concentration has reached the end of the chembers, the solution of Eq. (2.3-36) is best obtained by separation of variables. The result for the average concentration in the half cell is... 

Many partial differential equations arising in physical problems can be solved by separation of variables. In this procedure, a trial solution consisting of factors depending on one variable each is introduced, and the resulting equation is manipulated until the variables occur only in separate terms. Setting these terms equal to constants gives one ordinary differential equation for each variable. 

Recalling the method of solving such equations by separation of variables, you know that a solution of the first of these equations is A( ). B(2), and a solution of the second is A( 2). Z (l). Here A( 1), for example, denotes the Is wave function for hydrogen atom A, written for the coordinates of electron number 1 Since the Is wave function for hydrogen drops off exponentially with r, the wave function A( 1), in the coordinate system of Fig. 6.5, and with distances expressed in atomic units, is... 

The solution to this is obtained by separation of variables in a straightforward manner. 

This equation is an ordinary linear differential equation with constant coefficients if in and the rate constant are time independent. In this case the solution may be obtained by separation of variables and integration ... 

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