Variable separable differential equation

We shall neglect the parenthesis for r for brevity. The definition of the rate of reaction is a variable separable differential equation. In general, t may be computed from the following integral with appropriate initial conditions on c ... 

This is a variable separable differential equation. Integration with the appropriate boundary conditions gives... 

Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M x, y) dx + N x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = f f x) dx + c, which may or may not be expressible in simpler form. 

For a more complex scheme, one writes a separate differential equation for each variable. For Scheme I, Eq. (5-54), for example, the two rate equations are... 

Parabolic partial differential equations with homogenous boundary conditions are solved in this section. The dependent variable u is assumed to take the form u = XT, where X is a function of x alone and T is a function of t alone. This leads to separate differential equations for X and T. This methodology is best illustrated using an example. 

The task in these cases is to find a function that upon differentiation twice gives some combination of the same function, its derivative and variable. Such differential equations cannot be solved by the above separation of variables technique. A detailed discussion of how to solve second-order differential equations is beyond the scope of this book, but we will consider two special cases that often arise in computational chemistry. 

In order to employ the method of separation of variables, the differential equation and boimdary conditions are required to be linear and homogeneous. Presently, the linearity requirement is satisfied, but the boundary conditions are not homogeneous. In this problem, recasting the variables into dimensionless quantities reduces the problem to a homogeneous one. That is, let... 

Establish directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a x, x ) = the self-similar solution il/ rj) = e. (Hint Recognize the convolution on the right-hand side of (5.2.16). Letting = ij/ where ij/ is the derivative of the Laplace transform ij o ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to ij/). 

The equation just given is a first-order, separable differential equation that can be solved by separating the variables and integrating ... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

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. 

The variables are separable, but an integration in closed form is not possible because of the odd exponent. Numerical integration followed by substitution into (4) will provide both A and B as functions of t. The plots, however, are of solutions of the original differential equations with ODE. 

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. 

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... 

This differential equation is readily solved by separation of variables, leading to... 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

Equation (2.28) is now separable into two independent differential equations, one for each of the two independent variables x and t. The time-dependent equation is... 

The first term on the left-hand side of equation (2.77) depends only on the variable x, the second only on y, and the third only on z. No matter what the values of x, or y, or z, the sum of these three terms is always equal to the same constant E. The only way that this condition can be met is for each of the three terms to equal some constant, say E, Ey, and E, respectively. The partial differential equation (2.77) can then be separated into three equations, one for each variable... 

This partial differential equation may be readily separated by writing the wave function (R, r) as the product of two functions, one a function only of the center of mass variables X, Y, Z and the other a function only of the relative coordinates x, y, z... 

The left-hand side of equation (G.21) depends only on the variable 9, while the right-hand side depends only on cp. Following the same argument used in the solution of equation (2.28), each side of equation (G.21) must be equal to a constant, which we write as m. Thus, equation (G.21) separates into two differential equations... 

A first-order differential equation can always be solved, although its solution is not necessarily easy to obtain. If the variables are separable, the equation can be reduced to the form... 

Although the title of this chapter is general, it will be concerned only with the most important examples of partial differential equations of interest to physicists and chemists. Fortunately, the equations involved in virtually all of these applications can be solved by the very powerful method of separation of variables. 

A partial differential equation is one with two or more independent variables. The separation of these variables, if it can be carried out, yields ordinary differential equations which can, in most cases, be solved by the various methods presented in Chapters 3 and 5. The general approach to this problem will now be illustrated by a number of examples that are fundamental in physics and chemistry. 

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

The wave equation is a second-order partial differential equation in three variables. The usual technique for solving such an equation is to use a procedure known as the separation of variables. However, with r expressed as the square root of the sum of the squares of the three variables, it is impossible... 

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... 

In this book PDEs appear primarily in Section 8.1 and problem section P8.01. Some simpler methods of solution are mentioned there Separation of variables, application of finite differences and method of lines. Analytical solutions can be made of some idealized cases, usually in terms of infinite series, but the main emphasis in this area is on numerical procedures. Beyond the brief statements in Chapter 8, this material is outside the range of this book. Further examples are treated by WALAS (Modeling with Differential Equations in Chemical Engineering, 1991). 

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