# Fourier Series And Boundary Value Problems Churchill Brown Pdf

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College Physics — Raymond A. Serway, Chris Vuille — 8th Edition.

## Fourier Series and Boundary Value Problems Churchill and Brown - Mc Graw Hill

It is designed for students who have completed a first course in ordinary differential equations and the equivalent of a term of advanced calculus. In order that the book be accessible to as great a variety of students as possible, there are footnotes referring to texts which give proofs of the more delicate results in advanced calculus that are occasionally needed.

The physical applications, explained in some detail, are kept on a fairly elementary level. The first objective of the book is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. Representations of functions by Fourier series, involving sine and cosine functions, are given special attention. Fourier integral representations and expansions in series of Bessel functions and Legendre polynomials are also treated.

The second objective is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations. Some attention is given to the verification of solutions and to uniqueness of solutions; for the method cannot be presented properly without such considerations. Other methods are treated in the authors' book Complex Variables and Applications and in Professor Churchill's book Operational Mathematics.

This book is a revision of the edition. The first two editions, published in and , were written by Professor Churchill alone. While improvements appearing in earlier editions have been retained with this one, there are a number of major changes in this edition that should be mentioned.

The introduction of orthonormal sets of functions is now blended in with the treatment of Fourier series. Orthonormal sets are thus instilled earlier and are reinforced immediately with available examples. Also, much more attention is now paid to solving boundary value problems involving nonhomogeneous partial differential equations, as well as problems whose nonhomogeneous boundary conditions prevent direct application of the method of separation of variables. To be specific, considerable use is made, both in examples and in problem sets, of the method of variation of parameters, where the coefficients in xv xvi PREFACE certain eigenfunction expansions are found by solving ordinary differential equations.

Other improvements include a simpler derivation of the heat equation that does not involve vector calculus, a new section devoted exclusively to examples of eigenfunction expansions, and many more figures and problems to be worked out by the reader. There has been some rearrangement of the early material on separation of variables, and the exposition has been improved throughout. The chapters on Bessel functions and Legendre polynomials, Chapters 7 and 8, are essentially independent of each other and can be taken up in either order.

The last three sections of Chapter 2, on further properties of Fourier series, and Chapter 9, on uniqueness of solutions, can be omitted to shorten the course; this also applies to some sections of other chapters. The preparation of this edition has benefited from the continued interest of various people, many of whom are colleagues and students.

Representations by series are encountered in solving such boundary value problems. The theories of those representations can be presented independently. They have such attractive features as the extension of concepts of geometry, vector analysis, and algebra into the field of mathematical analysis. Their mathematical precision is also pleasing. But they gain in unity and interest when presented in connection with boundary value problems. The set of functions that make up the terms in the series representation is determined by the boundary value problem.

Representations by Fourier series, which are certain types of series of sine and cosine functions, are associated with a large and important class of boundary value problems. We shall give special attention to the theory and application of Fourier series. But we shall also consider extensions and generalizations of such series, concentrating on Fourier integrals and series of Bessel functions and Legendre polynomials.

I A boundary value problem is correctly set if it has one and only one solution within a given class of functions. Physical interpretations often suggest boundary conditions under which a problem may be correctly set. In fact, it is sometimes helpful to interpret a problem physically in order to judge whether the boundary conditions may be adequate.

This is a prominent reason for associating such problems with their physical applications, aside from the opportunity to illustrate connections between mathematical analysis and the physical sciences. The theory of partial differential equations gives results on the existence and uniqueness of solutions of boundary value problems. But such results are necessarily limited and complicated by the great variety of types of differential equations and domains on which they are defined, as well as types of boundary conditions.

Instead of appealing to general theory in treating a specific problem, our approach will be to actually find a solution, which can often be verified and shown to be the only one possible. The equations that represent those boundary conditions may involve values of derivatives of u, as well as values of u itself, at points on the boundary.

In addition, some conditions on the continuity of u and its derivatives within the domain and on the boundary may be required. Such a set of requirements constitutes a boundary value problem in the function u. We use that terminology whenever the differential equation is accompanied by some boundary conditions, even though the conditions may not be adequate to ensure the existence of a unique solution of the problem.

Frequently, it is convenient to indicate partial differentiation by writing independent variables as subscripts. The differential equation is defined in the first quadrant of the xy plane. The function 5 and its partial derivatives of the first and second order are continuous in the region x 0, y 0. A differential equation in a function u, or a boundary condition on u, is linear if it is an equation of the first degree in u and derivatives of u.

Thus the terms of the equation are either prescribed functions of the independent variables alone, including constants, or such functions multiplied by u or a derivative of u.

The boundary value problems in Examples 1 and 2 are, therefore, linear. The method of solution presented in this book does not apply to nonlinear problems. A linear differential equation or boundary condition in u is homogeneous if each of its terms, other than zero itself, is of the first degree in the function u and its derivatives. Homogeneity will play a central role in our treatment of linear boundary value problems. Observe that equation 3 and the first of conditions 4 are homogeneous but that the second of those conditions is not.

Equation 6 is homogeneous in a domain of the xy plane only when the function G is identically zero G 0 throughout that domain; and equation 7 is nonhomogeneous unless f y, z 0 for all values of y and z being considered.

It is convenient to refer to that transfer as a flow of heat, as if heat were a fluid or gas that diffused through the body from regions of high concentration into regions of low concentration. Let P0 denote a point x0, y0, z0 interior to the body and S a plane or smooth curved surface through P0.

Also, let n be a unit vector that is normal to S at the point P0 Fig. At time t, the flux ofheat t x0, y0, z0, t across S at in the direction of n is the quantity of heat per unit area per unit time that is being conducted across S at P0 in that direction. Flux is, therefore, measured in such units as calories per square centimeter per second. This is the quantity of heat required to raise the temperature of a unit mass of the material one unit on the temperature scale.

Unless otherwise stated, we shall always assume that the coefficients K and a are constants and that the same is true of 6, the mass per unit volume of the material. With these assumptions, a second postulate in the mathematical theory is that conduction leads to a temperature function u which, together with its derivative and those of the first and second order with respect to x, y, and z, is continuous throughout each domain interior to a solid body in which no heat is generated or lost.

Suppose now that heat flows only parallel to the x axis in the body, so that flux and temperatures U depend on only x and t. We then construct a small rectangular parallelepiped, lying in the interior of the body, with one vertex at a point x, y, z and with faces parallel to the coordinate planes. The lengths of the edges are Ax, Ay, and Az, as shown in Fig. Observe that, since the parallelepiped is small, the continuous function Ut varies little in that region and has approximately the value it throughout it.

This approximation improves, of course, as A x tends to zero. The mass of the element of material occupying the parallelepiped is 8 Ax Ay Az. So, in view of the definition of specific heat a stated above, we know that one measure of the quantity of heat entering that element per unit time at time t is approximately 2 Ax Ltsy t.

Another way to measure that quantity is to observe that, since the flow of heat is parallel to the x axis, heat crosses only the surfaces ABCD and EFGH of the element, which are parallel to the yz plane. In the derivation of equation 4 , we assumed that there is no source or sink of heat within the solid body, but only heat transfer by conduction. The rate Q per unit volume at which heat is generated may, in fact, be any continuous function of x and t, in which case the term q in equation 5 also has that property.

The heat equation describing flow in two and three dimensions is cussed in Sec. By considering the rate of heat passing through each of the six faces of the element in Fig. The derivation of equation 1 in Problem 6, Sec. Equations that describe thermal conditions on the surfaces of the solid body and initial temperatures throughout the body must accompany the heat equation if we are to determine the temperature function u.

The conditions on the surfaces may be other than just prescribed temperatures. Suppose, for example, that the flux F into the solid at points on a surface S is some constant That is, at each point P on 5, units of heat per unit area per unit time flow across S in the opposite direction of an outward unit normal vector n at P. From Fourier's law 1 in Sec. Hence du 8 on the surface S. On the other hand, there may be surface heat transfer between a boundary surface and a medium whose temperature is a constant T.

The inward flux which can be negative, may then vary from point to point on 5; and we assume that, at each point P, the flux is proportional to the difference between the temperature of the medium and the temperature at P. Consider a semi-infinite slab occupying the region 0 c x c c, y 0 of three-dimensional space. Figure 3 shows the cross section of the slab in the xy plane.

It should be emphasized that the various partial differential equations in this section are important in other areas of applied mathematics. In that case, ct' represents the mass of the substance that is diffused per unit area per unit time through a surface, u denotes concentration the mass of the diffusing substance per unit volume of the solid , and K is the coefficient of diffusion. Since the mass of the substance entering the element of volume in Fig.

We have seen in this section that the steady-state temperatures at points interior to a solid body in which no heat is generated are represented by a harmonic function.

The steady-state concentration of a diffusing substance is also represented by such a function. Among the many physical examples of harmonic functions, the velocity potential for the steady-state irrotational motion of an incompressible fluid is prominent in hydrodynamics and aerodynamics.

## Fourier Series and Boundary Value Problems

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College Physics — Raymond A. Serway, Chris Vuille — 8th Edition. Introduction to Heat Transfer — Frank P. Incropera — 6th Edition. Nixon, Alberto S.

### Fourier Series and Boundary Value Problems Churchill and Brown - Mc Graw Hill

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M,F , A. The Math Library CourseLib page. Churchill, Mcgraw-Hill, New York, 7th edition. Topics include: Orthonormal functions, best approximation in the mean, Fourier series, convergence pointwise and in the mean. Applications to boundary value problems, Sturm-Liouville equations, eigenfunctions, Fourier transform and applications.

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