Separation of variables

Consider a one-dimensional diffusion equation of the following form  

The method of separation of variables is applicable for linear homogeneous equations, i.e., C(u + v) = Cu + Cv, C(au) = aCu, in a finite domain. The method of separation of variables assumes that 

Because the equation is linear, the principle of superposition applies, i.e., the sum of linearly independent solutions is also a solution. Thus a general solution is 

The expression for the coefficient A can be obtained as follows. The initial condition gives 

Multiplying the above equation by sin(mTtx) and integrating over (0,1) gives 

A frequent case is that the wave function depends on two independent variables, x, and X2, and that the Hamiltonian can be written as a sum  

Quite generally, a function of two variables may be expanded in a series of the type 

If Equation 1.42 is divided by q (x,)x(x2), we obtain, after some calculation 

The left member of Equation 1.43 only depends on x while the right member only depends on Xj. The right and the left members may thus be varied independently of each other therefore Equation 1.43 is valid only if both members are equal to a constant K. Hence, 

We have shown that both cp and x are eigenfunctions of Hj and H2, respectively. 

Let us use the following equation and solve the equation using the separating variables method. 

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EXAMPLE 4-5 Anodalplane is one through which a wavefunction is antisymmetric for reflection. Consider the Mxy and Myz orbitals of Fig. 4-8. Through which planes do these two orbitals have different reflection symmetries  

SOLUTION idxy is antisymmetric for reflection only throngh the x, z and y, z planes. 3dyz is antisymmetric for reflection only through the x, z and x, y planes. Hence, these orbitals differ in their symmetries for reflection through the y, z and x, y planes. 

We shall indicate in some detail the way in which the Schrodinger equation (4-6) is solved. Recall the strategy of separating variables which we used in Section 2-7  

Express as a product of functions, each depending on only one variable. 

Substitute this product into the Schriklinger equation and try to manipulate it so that the equation becomes a sum of terms, each depending on a single variable. These terms must sum to a constant. 

Recall that ODEs generally yield solutions containing one or more arbitrary constants, which can be determined from boundary conditions. In contrast, solutions of PDEs often contain arbitrary functions. An extreme case is the second-order equation 

The simplest way to solve PDEs is to reduce a PDE with n independent variables to n independent ODEs, each depending on just one variable. This is not always possible, but we will limit our consideration to such cases. In the preceding section, we were able to reduce the wave equation and the heat equation to the Helmholtz equation when the time dependence of T(r, t) was separable. Consider the Helmholtz equation in Cartesian coordinates 

Chapter 12 Partial Differential Equations and Special Functions 

If the boundary conditions allow, the solution might be reducible to a separable function of x, y, z  

if we solve for the term X (x)/X(x), we find that this function of jc alone must be equal to a function of y and z, for arbitrary values of jc, y, z. The only way this is possible is for the function to equal a constant, for example. 

As discussed in the previous section, the problem is solving a differential equation with respect to either the position (classical) or wave function (quantum) for the particles in the system. The standard method of solving differential equations is to find a set of coordinates where the differential equation can be separated into less complicated equations. The first step is to introduce a centre of mass coordinate system, defined as the mass-weighted sum of the coordinates of all particles, which allows the translation 

If an approximate separation is not possible, the many-body problem can often be transformed into a pseudo one-particle system by taking the average interaction into account. For quantum mechanics, this corresponds to the Hartree-Fock approximation, where the average electron-electron repulsion is incorporated. Such pseudo one-particle solutions often form the conceptual understanding of the system, and provide the basis for more refined computational methods. 

Molecules are sufficiently heavy that their motions can be described quite accurately by classical mechanics. In condensed phases (solution or solid state), there is a strong interaction between molecules, and a reasonable description can only be attained by having a large number of individual molecules moving under the influence of each other s repulsive and attractive forces. The forces in this case are complex and cannot be written in a simple form such as the Coulomb or gravitational interaction. No analytical solutions can be found in this case, even for a two-particle (molecular) system. Similarly, no approximate solution corresponding to a Hartree-Fock model can be constructed. The only method in this case is direct simulation of the full dynamical equation. 

The spherical polar expression for the Laplacian operator appears much more foreboding than the Cartesian coordinate version. However, this is not really the case, since now the Schrddinger equation can be split into radial and angular equations that can be solved separately. To see this, we first write the wavefunction as a product of a function that only depends on r and a function that only depends on the angles 9 and  

The first function, Rp r), will control the behaviour of the function on any line moving directly away from the nuclear centre, while the second gives the angular behaviour as we move around the nucleus. The Schrodinger equation for the H atom problem can now be written  

Notice that in the first set of terms the angular function has been cancelled when we divide by RpYp but the radial function siuwived. This is because the angular function is not affected by the radial operators and so can be written to the left of them and cancelled. The radial function cannot be moved so easily, because it will be affected by the differential operators in the first term. A similar rearrangement has been used to cancel Rp from the angular term. 

In the sections that follow, solution techniques for linear boundary value problems are developed. Specifically, the method of separation of variables in Section 6.2 is illustrated. In Section 6.3, the method of eigenfunction expansion is outlined. In Section 6.4, the method of Laplace transform is illustrated. The method of combination of variables is outlined in Section 6.5. In Section 

Each method is accompanied by at least two worked-out examples demonstrating the important steps in the construction of a solution to a given problem. Also, the method of regular perturbation, a technique that can be very helpful in estimating a solution to some nonlinear problems, is briefly introduced in this chapter. 

Separation of variables is one of the most widely used solution techniques for a system consisting of a second-order PDE with boundary and/or initial conditions. This solution technique requires that the PDE be reduced to a 

The time or time-like part of the PDE usually results in an initial value type problem. The general solution of this initial value problem is then combined with the eigenfunctions resulting from solving the boundary value problem. Application of the initial condition usually results in a Eourier series. 

In order for this technique to be successfully applied, both the PDE and accompanying boundary conditions must be linear and homogeneous. To demonstrate the technique, we will consider the following model  

In the two-stream approximation the continuous radiation field is replaced by one traveling in only two directions. Hence the integral in Eq. (2.5.10) is replaced by a sum, 

In the thermal infrared the radiation field tends to be axially symmetric because no off-axis localized sources (such as the Sun) contribute significantly. Thus, only the azimuth-independent (m = 0) transfer equation is required. We have, from Eqs. (2.5.10) through (2.5.13), 

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Four volumetric defects are also included a spherical cavity, a sphere of a different material, a spheroidal cavity and a cylinderical cavity (a side-drilled hole). Except for the spheroid, the scattering problems are solved exactly by separation-of-variables. The spheroid (a cigar- or oblate-shaped defect) is solved by the null field approach and this limits the radio between the two axes to be smaller than five. 

In cases where the elassieal energy, and henee the quantum Hamiltonian, do not eontain terms that are explieitly time dependent (e.g., interaetions with time varying external eleetrie or magnetie fields would add to the above elassieal energy expression time dependent terms diseussed later in this text), the separations of variables teehniques ean be used to reduee the Sehrodinger equation to a time-independent equation. 

Beeause there are no terms in this equation that couple motion in the x and y directions (e.g., no terms of the form x yb or 3/3x 3/3y or x3/3y), separation of variables can be used to write / as a product /(x,y)=A(x)B(y). Substitution of this form into the Schrodinger equation, followed by collecting together all x-dependent and all y-dependent terms, gives ... 

A product wavefunetion is appropriate because the total Hamiltonian involves the kinetic plus potential energies of nine electrons. To the extent that this total energy can be represented as the sum of nine separate energies, one for each electron, the Hamiltonian allows a separation of variables... 

In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

Perform a separation of variables and indieate the general solution for the following expressions ... 

The separations of variables device assumes that /(x,y) can be written as a product of a function of x and a function of y ... 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Separation of Variables This is a powerful, well-utilized method which is applicable in certain circumstances. It consists of assuming that the solution for a partial differential equation has the form U =f x)g(y)- If it is then possible to obtain an ordinary differential equation on one side of the equation depending only on x and on the other side only on y, the partial differential equation is said to be separable in the variables x, y. If this is the case, one side of the equation is a function of x alone and the other of y alone. The two can be equal only if each is a constant, say X. Thus the problem has again been reduced to the solution of ordinaiy differential equations. 

In the case of the free jet, the solution for the Aaberg exhaust system can be found by solving the Laplace equation by the method of separation of variables and assuming that there is no fluid flow through the surface of the workbench. At the edge of the jet, which is assumed to be at 0—0, the stream function is given by Eq. (10.113). This gives rise to... 

A prominent part of many of the techniques is separation of variables. In that method, the deflection variables, or the variation In deflection variables, are arbitrarily separated into functions of plate coordinate x alone times functions of y alone. Wang [5-8] determined that separation of variables leads to exact solutions for some classes of plate problems, but does not for others, I.e., the deflections are not always separable. A specific example of an approximate use of separation of variables due to Ashton [5-9] will be discussed in Section 5.3.2. Other exact uses of the method abound throughout Section 5.3 through 5.5. 

The solution to the governing differential equation, Equation (5.32), is not as simple as for specially orthotropic laminated plates because of the presence of D. g and D2g. The Fourier expansion of the deflection w. Equation (5.29), is an example of separation of variables. However, because of the terms involving D.,g and D2g, the expansion does not satisfy the governing differential equation because the variables are not separable. Moreover, the deflection expansion also does not satisfy the boundary conditions. Equation (5.33), again because of the terms involving D. g and D2g. 

After separation of variables, Eq. (3-3) is integrated to give Eq. (3-4) as the integrated rate equation. 

Separation of variables and integration gives the indefinite integral, Eq. (3-8). 

If V is not a function of time, the Schrodinger equation can be simplified using the mathematical technique known as separation of variables. If we write the wavefunction as the product of a spatial function and a time function ... 

We need to investigate the conditions under which this is true, and to do this we make use of a technique called separation of variables . We substitute the product wavefunction (3.3) into (3.2) to give... 

In order to investigate whether the wavefunction can indeed be written in this way, we use the separation of variables technique and so write a wavefunction of the form... 

Separation of variables and using limits from Figure 2-55 gives... 

The separation of variables is used with suitable integrations, which result in the following equations ... 

Equation (8.4.3) is a linear first-order differential equation of concentration and reactor length. Using the separation of variables technique to integrate (8.4.3) yields... 

After substitution of k- = k K, rearrangement, separation of variables, and integration between the usual limits, we obtain... 

Since our Hamiltonian involves a sum of hett(i), which are only functions of the coordinates and momenta of a single electron, we can use separation of variables and reduce the problem to m identical one-electron problems... 

The search for eigenfunctions and eigenvalues in the example of the simplest difference problem. The method of separation of variables being involved in the apparatus of mathematical physics applies equelly well to difference problems. Employing this method enables one to split up an original problem with several independent variables into a series of more simpler problems with a smaller number of variables. As a rule, in this situation eigenvalue problems with respect to separate coordinates do arise. Difference problems can be solved in a quite similar manner. 

Results we are going to obtain here will be needed in the sequel because applying the method of separation of variables leads to problems just of the same type. In the next chapters we give various examples of employing this method for the discovery of stability and convergence of concrete difference schemes. 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator i. y = - -... 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

Stability with respect to the initial data. Let us investigate the stability of scheme (II) with homogeneous boundary conditions by the method of separation of variables. For this, we proceed as usual. This amounts to applying to scheme (II) with homogeneous boundary conditions the identities... 

To decide for yourself whether scheme (16) is stable with respect to the initial data, a first step is to evaluate the solution of problem (16a). This can be done using the method of separation of variables and deriving estimate (18) in the grid Z/2(o j)-norm ... 

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