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7 Green's Functions and Nonhomogeneous Problems 7 Green’s Functions and Nonhomogeneous Problems “The young theoretical physicists of a generation or two earlier subscribed to the belief that: If you haven’t done something important by age 30, you never will. Obviously, they were unfamiliar with the history of George Green, the miller of Nottingham.” Julian Schwinger (1918-1994) The wave equation, heat equation, and Laplace’s equation are typical homogeneous partial differential equations. They can be written in the form Lu(x) = 0, where L is a differential operator. For example, these equations can be written as ¶2 − c2r2 u = 0, ¶t2 ¶ − kr2 u = 0, ¶t r2u = 0.( 7.1) George Green (1793-1841), a British In this chapter we will explore solutions of nonhomogeneous partial dif- mathematical physicist who had little formal education and worked as a miller ferential equations, and a baker, published An Essay on Lu(x) = f (x), the Application of Mathematical Analysis to the Theories of Electricity and Mag- by seeking out the so-called Green’s function. The history of the Green’s netism in which he not only introduced function dates back to 1828, when George Green published work in which what is now known as Green’s func- 2 tion, but he also introduced potential he sought solutions of Poisson’s equation r u = f for the electric potential theory and Green’s Theorem in his stud- u defined inside a bounded volume with specified boundary conditions on ies of electricity and magnetism. Re- the surface of the volume. He introduced a function now identified as what cently his paper was posted at arXiv.org, arXiv:0807.0088. Riemann later coined the “Green’s function”. In this chapter we will derive the initial value Green’s function for ordinary differential equations. Later in the chapter we will return to boundary value Green’s functions and Green’s functions for partial differential equations. As a simple example, consider Poisson’s equation, r2u(r) = f (r). 226 partial differential equations Let Poisson’s equation hold inside a region W bounded by the surface ¶W nˆ as shown in Figure 7.1. This is the nonhomogeneous form of Laplace’s ¶W equation. The nonhomogeneous term, f (r), could represent a heat source in a steady-state problem or a charge distribution (source) in an electrostatic problem. Now think of the source as a point source in which we are interested in W the response of the system to this point source. If the point source is located at a point r0, then the response to the point source could be felt at points r. We will call this response G(r, r0). The response function would satisfy a Figure 7.1: Let Poisson’s equation hold point source equation of the form inside region W bounded by surface ¶W. r2G(r, r0) = d(r − r0). 0 The Dirac delta function satisfies Here d(r − r ) is the Dirac delta function, which we will consider in more d(r) = 0, r 6= 0, detail in Section 9.4. A key property of this generalized function is the Z sifting property, d(r) dV = 1. Z W d(r − r0) f (r) dV = f (r0). W The connection between the Green’s function and the solution to Pois- son’s equation can be found from Green’s second identity: Z Z [fry − yrf] · n dS = [fr2y − yr2f] dV. ¶W W 0 1 We note that in the following the vol- Letting f = u(r) and y = G(r, r ), we have1 ume and surface integrals and differen- r Z tiation using are performed using the [u(r)rG(r, r0) − G(r, r0)ru(r)] · n dS r-coordinates. ¶W Z h i = u(r)r2G(r, r0) − G(r, r0)r2u(r) dV W Z = u(r)d(r − r0) − G(r, r0) f (r) dV W Z = u(r0) − G(r, r0) f (r) dV.( 7.2) W Solving for u(r0), we have Z u(r0) = G(r, r0) f (r) dV W Z + [u(r)rG(r, r0) − G(r, r0)ru(r)] · n dS.( 7.3) ¶W If both u(r) and G(r, r0) satisfied Dirichlet conditions, u = 0 on ¶W, then the 2 In many applications there is a symme- last integral vanishes and we are left with2 try, ( 0) = ( 0 ) Z G r, r G r , r . u(r0) = G(r, r0) f (r) dV. Then, the result can be written as W Z u(r) = G(r, r0) f (r0) dV0. So, if we know the Green’s function, we can solve the nonhomogeneous W differential equation. In fact, we can use the Green’s function to solve non- homogenous boundary value and initial value problems. That is what we will see develop in this chapter as we explore nonhomogeneous problems in more detail. We will begin with the search for Green’s functions for ordi- nary differential equations. green’s functions and nonhomogeneous problems 227 7.1 Initial Value Green’s Functions In this section we will investigate the solution of initial value prob- lems involving nonhomogeneous differential equations using Green’s func- tions. Our goal is to solve the nonhomogeneous differential equation a(t)y00(t) + b(t)y0(t) + c(t)y(t) = f (t),( 7.4) subject to the initial conditions 0 y(0) = y0 y (0) = v0. Since we are interested in initial value problems, we will denote the inde- pendent variable as a time variable, t. Equation (7.4) can be written compactly as L[y] = f , where L is the differential operator d2 d L = a(t) + b(t) + c(t). dt2 dt The solution is formally given by y = L−1[ f ]. The inverse of a differential operator is an integral operator, which we seek to write in the form Z y(t) = G(t, t) f (t) dt. The function G(t, t) is referred to as the kernel of the integral operator and G(t, t) is called a Green’s function. is called the Green’s function. In the last section we solved nonhomogeneous equations like (7.4) using the Method of Variation of Parameters. Letting, yp(t) = c1(t)y1(t) + c2(t)y2(t),( 7.5) we found that we have to solve the system of equations 0 0 c1(t)y1(t) + c2(t)y2(t) = 0. f (t) c0 (t)y0 (t) + c0 (t)y0 (t) = .( 7.6) 1 1 2 2 q(t) This system is easily solved to give 0 f (t)y2(t) c1(t) = − 0 0 a(t) y1(t)y2(t) − y1(t)y2(t) 0 f (t)y1(t) c2(t) = 0 0 .( 7.7) a(t) y1(t)y2(t) − y1(t)y2(t) 228 partial differential equations We note that the denominator in these expressions involves the Wronskian of the solutions to the homogeneous problem, which is given by the deter- minant ( ) ( ) y1 t y2 t W(y1, y2)(t) = 0 0 . y1(t) y2(t) When y1(t) and y2(t) are linearly independent, then the Wronskian is not zero and we are guaranteed a solution to the above system. So, after an integration, we find the parameters as Z t f (t)y2(t) c1(t) = − dt t0 a(t)W(t) Z t f (t)y1(t) c2(t) = dt,( 7.8) t1 a(t)W(t) where t0 and t1 are arbitrary constants to be determined from the initial conditions. Therefore, the particular solution of (7.4) can be written as Z t Z t f (t)y1(t) f (t)y2(t) yp(t) = y2(t) dt − y1(t) dt.( 7.9) t1 a(t)W(t) t0 a(t)W(t) We begin with the particular solution (7.9) of the nonhomogeneous differ- ential equation (7.4). This can be combined with the general solution of the homogeneous problem to give the general solution of the nonhomogeneous differential equation: Z t Z t f (t)y1(t) f (t)y2(t) yp(t) = c1y1(t) + c2y2(t) + y2(t) dt − y1(t) dt. t1 a(t)W(t) t0 a(t)W(t) (7.10) However, an appropriate choice of t0 and t1 can be found so that we need not explicitly write out the solution to the homogeneous problem, c1y1(t) + c2y2(t). However, setting up the solution in this form will allow us to use t0 and t1 to determine particular solutions which satisfies certain homogeneous conditions. In particular, we will show that Equation (7.10) can be written in the form Z t y(t) = c1y1(t) + c2y2(t) + G(t, t) f (t) dt,( 7.11) 0 where the function G(t, t) will be identified as the Green’s function. The goal is to develop the Green’s function technique to solve the initial value problem 00 0 0 a(t)y (t) + b(t)y (t) + c(t)y(t) = f (t), y(0) = y0, y (0) = v0.( 7.12) We first note that we can solve this initial value problem by solving two separate initial value problems. We assume that the solution of the homo- geneous problem satisfies the original initial conditions: 00 0 0 a(t)yh (t) + b(t)yh(t) + c(t)yh(t) = 0, yh(0) = y0, yh(0) = v0.( 7.13) green’s functions and nonhomogeneous problems 229 We then assume that the particular solution satisfies the problem 00 0 0 a(t)yp (t) + b(t)yp(t) + c(t)yp(t) = f (t), yp(0) = 0, yp(0) = 0.( 7.14) Since the differential equation is linear, then we know that y(t) = yh(t) + yp(t) is a solution of the nonhomogeneous equation.
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