# FOURIER SERIES AND BOUNDARY VALUE PROBLEMS CHURCHILL PDF

Download Best Book Fourier Series and Boundary Value Problems (Brown and Churchill Series), PDF FILE Download Fourier Series and. Student's Solution Manual to Complex Variables and Applications 8th Ed. Fourier Series ( Edition) () by Georgi P. Tolstov. H. F. Weinberger - A First Course in Partial Differential Equations With Complex Variables and Transform Methods. Fourier series and boundary value problems / James Ward Brown, Ruel V. Ruel V. Churchill; Creator: Brown, James Ward; Subject: Fourier series Churchill, Ruel V. (Ruel Vance), ; Digital Description: application/pdf, xvi, p.

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Brown J.W., Churchill R.V. Fourier series and boundary value problems (5ed., and Boundary Value. Problems with Fourier Series 2e bw Pearson pdf. Fourier Series and Boundary Value Problems (Brown and Churchill) James Brown, Ruel Churchill and their applications to boundary value problems in partial differential equations of Brown, Ruel Churchill ebook PDF download. Fourier. Complex variables and applications / James Ward Brown, Ruel V. Churchill. coauthor with Dr. Churchill of Fourier Series and Boundary Value Problems, now.

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Incompletes will be given only in cases where the student has completed all but a small fraction of the course with a grade of C or better and a severe unexpected event prevents completion of the course. The last day to cancel, without permission from your College office, is the last day of the sixth week.

Homework assignments from the textbook, and from my own notes. The theory will predominate, but there will be considerable attention to applications in other fields. Notes, homework assignments, examples, practice exams, etc. Student Conduct: Statement on Scholastic Conduct: Each student should read the college bulletin for the definitions and possible penalties for scholastic dishonesty.

Students suspected of cheating will be reported to the Scholastic Conduct Committee. See our Privacy Policy and User Agreement for details. Published on Jul 23, Published by McGraw-Hill since its first edition in , this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics.

It will primarily be used by students with a background in ordinary differential equations and advanced calculus. There are two main objectives of this text. SlideShare Explore Search You. Submit Search. Successfully reported this slideshow.

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## Ruel Vance Churchill

Upcoming SlideShare. Like this presentation? Why not share! Bridge to Abstract Mathematics: Mathematical Proof and Structures Parzynski and Zipse: Introduction to Mathematical Analysis Pinsky: Differential Equations with Applications Rudin: Functional Analysis Rudin: Principles of Mathematical Analysis Rudin: Real and Complex Analysis Simmons: Elementary Number Theory Walker: Titles on the Current List Include: A complete list of Schaum titles may be obtained by writing to: New York St.

The editors were Richard H. Wallis, Maggie Lanzillo, and James W. Bradley; the production supervisor was Denise L.

The cover was designed by John Hite. All rights reserved. Copyright renewed by Rue! Printed in the United States of America. Except as permitted under the United States Copyright Act of , no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

Churchill's name appears first on the earlier editions. Includes bibliographical references and index. ISBN 1. Fourier series. Functions, Orthogonal. Boundary value problems. Churchill, Ruel Vance, date.

B76 '. He earned his A. He was coauthor with Dr. Churchill of Complex Variables and Applica.. Brown is listed in Who's Who in America. RUEL V. He received his B. Brown of Complex Variables and Applica- tions, a classic text that he first wrote over 45 years ago. He was also the author of Operational Mathematics, now in its third edition. Churchill held various offices in the Mathematical Association of America and in other mathematical societies and councils.

His fame is based on his mathematical theory of heat conduction, a theory involving expansions of arbitrary functions in certain types of trigonometric series. Although such expansions had been investigated earlier, they bear his name because of his major contributions. Fourier series are now fundamental tools in science, and this book is an introduction to their theory and applications. Fourier's life was varied and difficult at times. Orphaned by the age of 9, he became interested in mathematics at a military school run by the dictines in Auxerre.

He was an active supporter of the Revolution and narrowly escaped imprisonment and execution on more than one occasion. After the Revolution, Fourier accompanied Napoleon to Egypt in order to set up an educational institution in the newly conquered territory. Shortly after the French withdrew in , Napoleon appointed Fourier prefect of a department in southern France with headquarters in Grenoble.

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It was in Grenoble that Fourier did his most important scientific work. Since his professional life was almost equally divided between politics and science and since it was intimately geared to the Revolution and Napoleon, his advancement of the frontiers of mathematical science is quite remarkable. He died at the age of 62 on May 16, Conduction of Heat. Higher Dimensions and Boundary Conditions.

The Laplacian in Cylindrical and Spherical Coordinates. A Vibrating String. Vibrations of Bars and Membranes. Types of Equations and Boundary Conditions. Methods of Solution. On the Superposition of Separated Solutions. Inner Products and Orthonormal Sets. Generalized Fourier Series. Fourier Cosine Series. Fourier Sine Series. Fourier Series. Best Approximation in the Mean. One-Sided Derivatives. Two Lemmas. A Fourier Theorem. Discussion of the Theorem and Its Corollary.

Fourier Series on Other Intervals. Uniform Convergence of Fourier Series. Differentiation and Integration of Fourier Series. Convergence in the Mean.

Principle of Superposition. A Temperature Problem. Verification of Solution. A Vibrating String Problem. Historical Development. The Slab with Internally Generated Heat. Dirichlet Problems. Other Types of Boundary Conditions.

A String with Prescribed Initial Velocity. An Elastic Bar. Fourier Series in Two Variables. Periodic Boundary Conditions. Orthogonality of Eigenfunctions.

Uniqueness of Eigenfunctions. Examples of Eigenfunction Expansions. Surface Heat Transfer. Polar Coordinates. Modifications of the Method. A Vertically Hung Elastic Bar. An Integration Formula. A Fourier Integral Theorem. The Cosine and Sine Integrals. More on Superposition of Solutions. Temperatures in a Semi-Infinite Solid.

Temperatures in an Unlimited Medium. General Solutions of Bessel's Equation. Recurrence Relations. The Zeros of J0 x. Zeros of Related Functions. Orthogonal Sets of Bessel Functions. The Orthonormal Functions. Fourier-Bessel Series. Temperatures in a Long Cylinder. Heat Transfer at the Surface of the Cylinder.

Vibration of a Circular Membrane. Legendre Polynomials. Orthogonality of Legendre Polynomials. Rodrigues' Formula and Norms. Legendre Series. Dirichlet Problems in Spherical Regions. Steady Temperatures in a Hemisphere. Uniqueness of Solutions of the Heat Equation.

Solutions of Laplace's or Poisson's Equation. Solutions of a Wave Equation. 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 applica- tions, 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 representa- tions 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 Opera- tional 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.

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.

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They include Jacqueline R. Brown, Michael A. Lachance, Ronald P. Morash, Joyce A. Moss, Frank J. Papp, Richard L. Patterson, Mark A. Pinsky, and Sandra M. Ralph P. Boas, Jr. Brown furnished some of the references that are cited in the footnotes; and the derivation of the laplacian in spherical coordinates that is given was suggested by a note of R.

Agnew's in the American Mathematical Monthly, vol. Finally, it should be emphasized that this edition could not have been possible without the enthusiastic editorial support of people at McGraw-Hill, most especially Richard H. Wallis and Maggie Lanzillo. They, in turn, obtained the following reviewers of both the last edition and the present one in manuscript form: Joseph M.

Egar, Cleveland State University; K. Ryan, Northwestern State University of Louisiana. Representations by series are encountered in solving such boundary value problems. The theories of those representations can be presented indepen- dently. 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. 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 differential equations and boundary conditions in Examples I and 2 are all linear.

A boundary value problem is linear if its differential equation and all its boundary conditions are 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 consid- ered.

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. 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.

When the laplacian 2 is used, equation 1 takes the compact form 3 kV2U. The derivation of equation 1 in Problem 6, Sec. 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 bound- ary 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 diffus- ing 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.

An important harmonic func- tion in electrical field theory is the electrostatic potential V x, y, z in a region of space that is free of electric charges. The potential may be caused by a static distribution of electric charges outside that region. The fact that V is harmonic is a consequence of the inverse. Likewise, gravitational potential is a harmonic function in regions of space not occupied by matter.

In this book, the physical problems involving the laplacian, and Laplace's equation in particular, are limited mostly to those for which the differential equations are derived in this chapter.

Derivations of such differential equations in other areas of applied mathematics can be found in books on hydrodynamics, elasticity, vibrations and sound, electrical field theory, potential theory, and other branches of continuum mechanics.

A number of such books are listed in the Bibliography at the back of this book. I Often, because of the geometric configuration of the physical problem, it is more convenient to use the laplacian in other than rectangular coordinates. In this section, we show how the laplacian can be expressed in terms of the variables of two important coordinate systems already encountered in calculus. The cylindrical coordinates p, 4', and z determine a point P p, 4, z whose rectangular coordinates are Fig.

Thus p and 4, are the polar coordinates in the xy plane of the point Q, where Q is the projection of P onto that plane. Then, in view of relations 2 , it is also a function of the three independent variables p, 4, and z. If u is continuous and possesses continuous partial derivatives of the first and second orders, the following method, based on the chain rule for differentiating composite functions, can be used to express the laplacian 1 in terms of p, cb, and z.

The notation for spherical coordinates, treated later in this section, may also differ somewhat from that learned in calculus. The spherical coordinates r, 4, and 0 of a point P r, 4, 0 Fig.

This is accomplished in three steps, described below. First, we observe that, except for the names of the variables involved, transformation 16 is the same as transformation 2. A slab occupies the region 0 c x c c. Set up and solve the boundary value problem for the steady-state temperatures u x in the slab. Then obtain the expression hb2T the steady-state temperatures, where h is the ratio of the surface conductance H for to the thermal conductivity K of the material.

Derive the heat equation for those temperatures, where the constants k and q are the same ones as in equation 5 , Sec.

Modify the derivation of equation 5 , Sec. Since the faces are small, one may consider the needed flux at points on a given face to be constant over that face. A slender wire lies along the x axis, and surface heat transfer takes place along the wire into the surrounding medium at a fixed temperature T. Modify the procedure in Sec. Let r denote the radius of the wire, and apply Newton's law of cooling to see that the quantity of heat entering the element in Fig.

Suppose that the thermal coefficients K and a are functions of x, y, and z. Derive expressions 8 and 9 in Sec. In Sec. Use transformation 16 , Sec. Show that the physical dimensions of thermal conductivity k Sec. Let the tension of the string be great enough that the string behaves as if it were perfectly flexible. That is, at a point x, y on the string, the part of the string to the left of that point exerts a force T, in the tangential direction, on the part to the right; and any resistance to bending at the point is to be neglected.

The magnitude of the x component of the tensile force T is denoted by H. See Fig. Our final assumption here is that H is constant.

## Fourier Series And Boundary Value Problems

That is, the variation of H with respect to x and t can be neglected. They are adequately satisfied, for instance, by strings of musical instruments under ordinary conditions of operation. Mathematically, the as- sumptions will lead us to a partial differential equation in y x, it that is linear. Now let V x, it denote the y component of the tensile force T exerted by the left-hand portion of the string on the right-hand portion at the point x, y.

We take the positive sense of V to be that of the y axis. If cx is the angle of inclination of the string at the point x, y at time t, then —V x,t 1 H This is indicated in Fig. Equation 2 is also used in setting up certain types of boundary conditions. Suppose that all external forces such as the weight of the string and resistance forces, other than forces at the end points, can be neglected.

Consider a segment of the string not containing an end point and whose projection onto the x axis has length Ax. Since x components of displacements are negligible, the mass of the segment is 8 A x , where the constant 6 is the mass per unit length of the string. At time t, the y component of the force exerted by the string on the segment at the end x, y is V x, it , given by equation 2.

The tangential force S exerted on the other end of the segment by the part of the string to the right is also indicated in Fig. Now the acceleration of the end x, y in the y direction is it. When external forces parallel to the y axis act along the string, we let F denote the force per unit length of string, the positive sense of F being that of the y axis.

Then a term FAx must be added on the right-hand side of equation 4 , and the equation of motion is F 6 axis vertical and its positive sense upward, suppose that y the external force consists of the weight of the string.

If the In external force per unit length is a damping force proportional to the velocity in the y direction, for example, F is replaced by —Bye, where the positive constant B is a damping coefficient. Then the equation of motion is linear and homoge- neous: DOjOA U! OSflRD 1! SJfl0 0q1 'Jrq U! J2SUfl UOfl! OM Arm 03! E rn uo! A Jo uOfll? Z 'uogoanp fl?

P OAJflD 0q2 s3u! I that H is constant, regardless of what point or curve on the membrane is being discussed. In view of expressions 2 and 3 in Sec. Similar expressions are found for the vertical forces exerted over AD and BC when the tensile forces on those curves are considered.

From equation 7 , one can see that the static transverse displacements z x, y of a stretched membrane satisfy Laplace's equation Sec. Here the displacements are the result of displacements, perpendic- ular to the xy plane, of parts of the frame that support the membrane when no external forces are exerted except at the boundary.

A stretched string, with its ends fixed at the points 0 and 2c on the x axis, hangs at rest under its own weight. The y axis is directed vertically upward. Point out how it follows from the nonhomogeneous wave equation 7 , Sec. Use expression 2 , Sec. Thus show that Problem 3 is a membrane analogy for this temperature problem. Soap films have been used to display such analogies.

## Fourier Series and Boundary Value Problems

Give needed details in the derivation of equation 6 , Sec. The physical dimensions of H, the magnitude of the x component of the tensile force in a string, are those of mass times acceleration: A strand of wire 1 ft long, stretched between the origin and the point 1 on the x axis, weighs 0.

Assuming that no external forces act along the wire, state why the displace- ments y x, t should satisfy this boundary value problem: The bar is initially unstrained and at rest, with no external forces acting along it. Use expression 6 , Sec. Let z x, y denote the static transverse displacements in a membrane over which an external transverse force F x, y per unit area acts.

Show how it follows from the nonhomogeneous wave equation 8 , Sec. For each of these categories, equation 1 and its solutions have distinct features. Some indication of this is given in Problems 15 and 16, Sec. The terminology used here is suggested by the fact Problem 6, Sec. Hence it is elliptic throughout the xy plane. Poisson's equation Sec. Here v x, t represents either the electrostatic potential or current at time t at a point x units from one end of a transmission line or cable that has t A derivation of this equation is outlined in the book by Churchill , pp.

I electrostatic capacity K, self-inductance L, resistance R, and leakage conduc- tance S, all per unit length. It is parabolic if either K or L is zero. As indicated below, the three types of second-order linear equations just described require, in general, different types of boundary conditions in order to determine a solution.

Let u denote the dependent variable in a boundary value problem. A condition that prescribes the values of u itself along a portion of the boundary is known as a Dirichiet condition. The problem of determining a harmonic function on a domain such that the function assumes prescribed values over the entire boundary of that domain is called a Dirichiet problem.

In that case, the values of the function can be interpreted as steady-state temperatures. Such a physical interpretation leads us to expect that a Dirichlet problem may have a unique solution if the functions considered satisfy certain requirements as to their regularity.

Another type of boundary condition is a Robin condition. Initial values for both y and appear to be needed if the displacements y x, t are to be determined. This is suggested by interpreting u physically as a temperature function.

In this example, the boundary conditions are simple enough that we can actually determine the functions 4 and ift. Thesolution 12 of the boundary value problem consisting of equations 4 and 5 is known as d'Alembert's solution. It is easily verified under the assumption that f' x and f" x exist for all x. The method for solving boundary value problems illustrated in the two examples here has severe limitations.

But, even in the exceptional cases in which such general solutions can be found, the determination of the arbitrary functions directly from the boundary conditions is often difficult. Among a variety of other methods, the one to be developed in this book will be suggested by the example in Sec. That method, which is sometimes called the Fourier method, is a classical and powerful one. Before turning to it, however, we mention some other important ones.

Methods based on Laplace, Fourier, and other integral transforms, all included in the subject of operational mathematics, are especially effective. Even when a problem yields to more than one method, however, different methods sometimes produce different forms of the solution; and each form may have its own desirable features.

On the other hand, some problems require successive applications of two or more methods. Others, including some fairly simple ones, have defied all known exact methods. The development of new methods is an activity in present-day mathematical research. Namely, in seeking a solution of the boundary value problem in the example below, we shall find it necessary to expand an arbitrary function in a series of trigonometric functions Chap. The Dirichlet problem see Sec.

See the authors' book , also listed in the Bibliography. We shall not be concerned here with precise conditions on the function f. We assume only that f is bounded and observe that it is then physically reasonable to seek solutions of equation 1 that tend to zero as y tends to infinity.Bookmark it to easily review again before an exam.

Then use expressions 7 and 8 , Sec. Since the mass of the substance entering the element of volume in Fig. Since C1 and C2 cannot both vanish if X x is to be nontrivial, it follows that the positive number a must, in fact, be a positive integer n. The validity of the solution obtained will be established in Sec. Thus show that Problem 3 is a membrane analogy for this temperature problem.

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