PROBLEMS 513 16.2 Show that the Fourier transform of is given as 16.3 ing second-order inhomogeneous differential equation: Using the Laplace transform technique, find the solution of the follow- with the boundary conditions y(0) = 2 and y’(0) = -1. 16.4 Solve the following system of differential equations: 2z(t) - y(t) - y’(t) = 4(1 - exp(-t)) 2z’(t) +y(t) = 2(1 +3exp(-2t)) with the boundary conditions z(0) = y(0) = 0. 16.5 One end of an insulated semi-infinite rod is held at temperature T(t, 0) = To, with the initial conditions T(0, z) = 0 and T(t, m) = 0. Solve the heat transfer equation where k is the thermal conductivity, c is the heat capacity, and p is the density. 514 INTEGRAL TRANSFORMS Hint: The solution is given in terms of erf c as T(t,x) = Toerfc [IT] , where the erf c is defined in terms of erf as erf c(x) = 1 - erf x 16.6 t ion Find the current, I, for the IF1 circuit represented by differential equa- dI dt L- i- RI = E, with the initial condition I(0) = 0. E is the electromotive force and L, R, and E are constants. 16.7 of differential equations Using the Laplace transforms find the solution of the following system dx - +y = 3e2t dt -+z=O, dY dt subject to the initial conditions x(0) = 2, v(0) = 0. 16.8 Using the Fourier sine transform show the integral * s3sinsx e-xcosx=il ds, x > 0. 16.9 Using the Fourier cosine transform show the integral s2 + (cos sx)ds, x 2 0. 16.10 the end at the origin fixed. The shape of the string at t = 0 is given as Let a semi-infinite string be extended along the positive x-axis with Y(X, 0) = f(xL PROBLEMS 515 where y(x, t) represents the displacement of the string perpendicular to the x-axis and satisfies the wave equation a is a constant. - a2Y 2 a2Y at2 =a G’ Show that the solution is given as dscas(sat)sin(sz) 16.11 Establish the Fourier sine integral representation Hint: First show that 16.12 Show that the Fourier sine transform of xe-ax is given as 16.13 Establish the result 16.14 Use the convolution theorem to show that k?’ { (s2 fh2,.} = f cos ht + -sin 1 bt. 2h 16.15 of differentia1 equations: Use Laplace transforms to find the solution of the following system - -wy1 dYl dx - -a2yz - Q3Y3, dY3 dx 516 INTEGRAL TRANSFORMS with the boundary conditions yl(0) = CO, yz(0) = y3(0) = 0. 16.16 Laguerre polynomials satisfy zLK + (1 - t)L:, + nL,(z) = 0. Show that L'{L,(az)} = (s - s > 0. 17 VARIATIONAL ANALYSIS Variational analysis is basically the study of changes. We are often interested in finding how a system reacts to small changes in its parameters. It is for this reason that variational analysis hes found a wide range of applications not just in physics and engineering but also in finance and economics. In applications we frequently encounter caees where a physical property is represented by an intwal, the extremum of which is desired. Compared to ordinary calculus, where we deal with functions of numbers, these integrals are functions of some unknown function and its derivatives; thus, they are called functionals. Search for the extremum of a function yields the point at which the function is extremum. In the caee of functionals, variational analysis gives us a differential equation, which is to be solved for the extremizing function. After Newton’s formulation of mechanics mange developed a new ap pmch, where the equations of motion are obtained from a variational integral called action. This new formulation makes applications of Newton’s theory to many particles and continuous systems poesible. Today in making the transition to quantum mechanics and to quantum field theories wan formulation is a must. Geodesics are the shortest paths between two points in a given geametry and constitute one of the main applications of variational analysis. In Ein- stein’s theory of gravitation geodesics play a central role tu the paths of freely moving particles in curved spacetime. Variational techniques also form the mathematical basis for the finite elements method, which is a powerful tool for solving complex boundary value problems, and stability analysis. Variational analysis and the Rayleigh-Ritz method ale0 allows us to find approximate eijpnvalues and eigenfunctions of a Sturm-Liouville system. 517 518 VARIATIONAL ANALYSIS I I fig. 17.1 Variation of paths 17.1 PRESENCE OF ONE DEPENDENT AND ONE INDEPENDENT VARIABLE 17.1.1 Euler Equation A majority of the variational problems encountered in physics and engineering are expressed in terms of an integral: (17.1) where y(x) is the desired function and f is a known function depending on y, its derivative with respect to x, that is yx, and x. Because the unknown of this problem is a function, J is called a functional and we write it as J [Y (XI1 . (17.2) Usually the purpose of these problems is to find a function, which is a path in the xy-plane between the points (21, y1) and (zz, yz), which makes the func- tional J [y (z)] an extremum. In Figure 17.1 we have shown two potentially possible paths; actually, there are infinitely many such paths. The difference of these paths from the desired path is called the variation of y, and we show it as Sy. Because Sy depends on position, we use ~(z) for its position depen- dence and use a scalar parameter a as a measure of its magnitude. Paths close to the desired path can now be parametrized in terms of LY as Y(X, a) = y(x, 0) + ar](x) + O(O2), (17.3) PRESENCE OF ONE DEPENDENT AND ONE INDEPENDENT VARIABLE 519 where y(z, a = 0) is the desired path, which extremizes the functional J[Y(X)]. We can now express Sy as SY = Y(Z, a) - Y(Z,O) = arl(z) (17.4) and write J as a function of a as J(a) = L; f [Y(Z,~),YX(~,~),~ldX. (17.5) Now the extremum of J can be found as in ordinary calculus by impwing the condition (17.6) In this analysis we assume that T?(Z) is a differentiable function and the vari- ations at the end points are zero, that is, Now the derivative of J with respect to cy is Using equation (17.3) we can write and aYx(z, a) d?7(4 - aa & Substituting these in Equation (17.8) we obtain Integrating the second term by parts gives (17.7) (17.8) (17.9) (17.10) (17.11) (17.12) Using the fact that the variation at the end points are zero, we can write Equation (17.11) as (17.13) 520 VARlAT/ONAL ANALYSIS Because the variation ~(x) is arbitrary, the only way to satisfy this equation is by setting the expression inside the brackets to zero, that is, (17.14) In conclusion, variational analysis has given us a second-order differential equation to be solved for the path that extremizes the functional J[y(z)]. This differential equation is called the Euler equation. 17.1.2 To find another version of the Euler equation we write the total derivative of the function f (y, y,, x) as Another Form of the Euler Equation df - 8.f af dYx af -y, + + dx 8y ay, dx dx -_ Using the Euler equation [Eq. (17.14)] we write af d af ay dx a~, - = and substitute into Equation (17.15) to get This can also be written as f-yx- =o. af d [ ax dx (17.15) (17.16) (17.17) (17.18) This is another version of the Euler equation, which is extremely useful when f (y, yx, x) does not depend on the independent variable x explicitly. In such cases we can immediately write the first integral as 8.f aYx f - yx- = constant, (17.19) which reduces the problem to the solution of a first-order differential equation. 17.1.3 Example 17.1. Shortest path between two points: To find the shortest Applications of the Euler Equation path between two points in two dimensions we ds = [(dx)' + (d~)~]' write the line element as (17.20) PRESENCE OF ONE DEPENDENT AND ONE INDEPENDENT VARIABLE 521 The distance between two points is now given as a functional of the path and in terms of the integral (17.21) = l: [I +YE]' dx. (17.22) To find the shortest path we must solve the Euler equation for Because f(y, yz, x) does not depend on the independent variable ex- plicitly, we use the second form of the Euler equation [Eq. (17.19)] to write (17.23) where c is a constant. This is a first-order differential equation for y(x), and its solution can easily be found as y = ax + b. ( 17.24) This is the equation of a straight line, where the integration constants a and b are to be determined from the coordinates of the end points. The shortest paths between two points in a given geometry are called geodesics. Geodesics in spacetime play a crucial role in Einstein's theory of gravitation as the paths of free particles. Example 17.2. Shape of a soap film between two'rings: Let us find the shape of a soap film between two rings separated by a distance of 2x0. Rings pass through the points (x1,yl) and (x2,yz) as shown in Figure 17.2. Ignoring gravitation, the shape of the film is a surface of revolu- tion; thus it is sufficient to find the equation of a curve, y(x), between two points (21, y1) and (x2, yz). Because the energy of a soap film is proportional to its surface area, y(x) should be the one that makes the area a minimum. We write the infinitesimal area element of the soap film as dA = 2nyds = 2ny [l + y:]' dx. The area, aside from a factor of 2n, is given by the integral (17.25) ( 17.26) (17.27) 522 VARIATIONAL ANALYSIS fig. 17.2 Soap film between two circles Since f(y, yz, z) is given as f(Y,Yz,x) = Y [1 +Yp 1 which does not depend on x explicitly, we write the Euler equation as (17.28) = c1, [l + y3p (17.29) where c1 is a constant. Taking the square of both sides we write Y2 [l + y?] - + This leads us to the first-order differential equation (17.30) (17.31) which on integration gives x = c1 cash-’ + c2. (17.32) c1 Thus the function y(x) is determined as x - c;? y(z) = cl cosh (7) . (17.33) [...]... then the quantity, H , defined as dL H(qa, -, t ) = 8% c qi- 8L 8% -L is conserved Using Cartesian coordinates, interpret H 17.11 The brachistochrone problem: Find the shape of the curve joining two points, along which a particle, initially a t rest, falls freely under the influence of gravity from the higher point to the lower point in the least amount of time 17.12 In an expanding flat universe the... have no choice but to use integral equations In this chapter we discuss the basic properties of linear integral equations and introduce some techniques for obtaining their solutions We also discuss 54 7 548 INTEGRAL EQUATlONS the Hilbert-Schmidt theory, where an eigenvalue problem is defined in terms of linear integral operators 18.1 CLASSIFICATION OF INTEGRAL EQUATIONS Linear integral equations are classified... Volterra equations also have the following kin& a#O kind1 =1 kind I1 cr = a(%) kind I11 cy When F ( z ) is zero, the integral equation is called homogeneous Integral equations can also be defined in higher ciimen%ions In two dimensions we can write a linear integral equation as 18.2 INTEGRAL AND DIFFERENTIAL EQUATIONS Some integral equations can be obtained from differential equations To see this connection... o(a2), i = 1,2, a .,n, (17.41) where a is again a small parameter and the functions qi(x)are independent of each other We again take the variation at the end points as zero: rli(Z1) = 774x2)= 0 Taking the derivative of J ( a ) with respect to Q (17.42) and setting a to zero we get (17.43) Integrating the second term by parts and using the fact that a t the end points variations are zero, we write Equation... function is defined as (17.89) h=f+Ay Differentiating with respect to these parameters and integrating by parts and using the boundary conditions we get Taking the variations as (17.91) and using Equation (17.87) we write qj(z)dz=O, j = 1,2 (17.92) Because the variations q j are arbitrary, we set the quantity inside the square brackets to zero and obtain the differential equation dh dy d ah = 0 dxdy’... 17.10 Loaded cable fixed between two points: We consider a cable fixed between the points (0,O) and (1, h) The cable carries a load along the y-axis distributed as (17 .156 ) To find the shape of this cable we have to solve the variational problem (17 .157 ) with the boundary conditions y(0) = 0 and y(1) = h, (17 .158 ) where TOand qo are constants Using the Rayleigh-Ritz method we choose &(x) such that it satisfies... Sometimes in engineering problems we encounter functionals given as (17.66) where y(") stands for the n t h order derivative, the independent variable x takes values in the closed interval [a,b], and the dependent variable y ( z ) satisfies t h e boundary conditions d 1 (n1) y = Yo, ! / ' ( a ) = Y&, , y ( n - l ) ( a )= yo d b ) = Y1, Y'(b) = Yi, , y ( n - l ) ( b )= &-') , (17.67) Using the same method. .. conditions Integral equations, on the other hand, constitute a complete descrip tion of a given problem, where extra conditions are neither needed nor could be imposed Because the boundary conditions can be viewed as a convenient way of including global effects into a system, a connection between differential and integral equations is to be expected In fact, under certain conditions integral and differential... applications to many particle systems and continuous systems possible It is also a must in making the transition to quantum mechanics and quantum field theories For continuous systems we define a Lagrangian density L as L= Ld3?;’, (17.119) where V is the volume Now, the action in Hamilton’s principle becomes I = 6 Ldt (17.120) For a continuous timedependent system with n independent fields, &(T’,) , t a... possible to write theories that do not follow from a variational principle, they eventually run into problems We have seen that solving the Laplace equation within a volume V is equivalent to extremizing the functional with the appropriate boundary conditions Another frequently encountered differential equation in science and engineering is the Sturm-Liouville equation dx - ~(Z)U(.) + X ~ ( X ) U ( . applications not just in physics and engineering but also in finance and economics. In applications we frequently encounter caees where a physical property is represented by an intwal, the extremum. ANALYSIS In integral form this becomes (17.88) where the h function is defined as h=f+Ay. (17.89) Differentiating with respect to these parameters and integrating by parts and using the. from the coordinates of the end points. The shortest paths between two points in a given geometry are called geodesics. Geodesics in spacetime play a crucial role in Einstein's theory