THE SEPARATION OF VARIABLES METHOD FOR SECOND ORDER LINEAR PARTIAL DIFFERENTIAL EQUATIONS A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By Jorge Dimas Granados Del Cid June 2016 c 2016 Jorge Dimas Granados Del Cid ALL RIGHTS RESERVED ii The thesis of Jorge Dimas Granados Del Cid is approved. Borislava Gutarts, Ph.D., Committee Chair Anthony Shaheen, Ph.D. Kristin Webster, Ph.D. Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2016 iii ABSTRACT The Separation of Variables Method for Second Order Linear Partial Differential Equations By Jorge Dimas Granados Del Cid This thesis provides an overview of various partial differential equations, in- cluding their applications, classifications, and methods of solving them. We show the reduction (change of variables process) of an elliptic equation to the Laplace equa- tion (with lower order terms), as well as other cases. We derive the solutions of some partial differential equations of 2nd order using the method of separation of variables. The derivation includes various boundary conditions: Dirichlet, Neumann, mixed, periodic and Robin. A discussion of the eigenvalues related to various bound- ary conditions is provided. A discussion of Fourier series, as they apply to computing the coefficients of the series solutions, is included. The thesis concludes with a pre- sentation of open problems related to the topic. iv TABLE OF CONTENTS Abstract................................................................................................................. iv List of Figures........................................................................................................ vii Chapter 1. Introduction ..............................................................................................1 2. Reduction to Canonical Form ...................................................................6 2.1. Chain rule with respect to change of variables ...............................8 2.2. Hyperbolic reduction ...................................................................... 11 2.3. 2nd Hyperbolic Case........................................................................ 16 2.4. Parabolic Reduction ....................................................................... 18 2.5. Elliptic Canonical Reduction.......................................................... 21 3. Separation Of Variables ............................................................................ 27 3.1. Examples of second-order PDEs and Boundary Conditions .......... 27 3.2. The characteristic polynomial......................................................... 28 3.3. Dirichlet boundary conditions ........................................................ 30 3.4. The Heat Equation with Dirichlet Boundaries ............................... 32 3.5. Neumann Boundary Conditions ..................................................... 36 3.6. The Robin boundary conditions ..................................................... 40 4. Eigenvalues................................................................................................ 46 4.1. Eigenvalues: Dirichlet Boundary Conditions .................................. 46 4.2. Eigenvalues: Neumann Boundary Conditions................................. 48 4.3. Eigenvalues: The Robin's Boundary Conditions ............................ 49 5. Coefficients................................................................................................ 53 v 5.1. Coefficients in the Case of Dirichlet Boundary Conditions............. 53 5.2. Coefficients in the Case of Neumann Boundary Conditions ........... 56 vi LIST OF FIGURES Figure 1. Chain rule ................................................................................................6 2. sin(nπx=l) n = 1; 2; 3; 4: 0 < x < l: ..................................................... 32 3. cosh γl 6= 0................................................................................................ 39 4. a0 > 0; al > 0, eigenvalues as intersections............................................... 41 (−a0 + al) β 5. a0 < 0; al > 0 and a0 + al > 0, tan βl = 2 : ............................. 43 β + ala0 β 6. graph of − a = tan βl; where a > 0 .......................................................... 44 β 7. graph of − a = tan βl; where a < 0 .......................................................... 44 8. a0 = al = 1:9, l = π, β1 ≈ 0:758263778 ................................................... 45 9. for a0 > 0; al > 0, no negative eigenvalue ............................................... 51 10. for a0 < 0; al > 0, a0 + al > 0, no negative eigenvalue. .......................... 51 11. for a0 < 0; a0 + a1 < −a0all, one negative eigenvalue where the func- tions intersect. ........................................................................................ 52 vii CHAPTER 1 Introduction A partial differential equation (PDE) is an equality composed of mathematical entities that include an unidentified, multivariable function and its partial derivatives. Definition 1.1. A partial derivative is the derivative with respect to one variable of a function of several variables, with the remaining variables treated as constants. For instance, the partial derivative of u with respect to x is u(x + h; y; z; : : :) − u(x; y; z; : : :) lim h!0 h These partial derivatives of u, with respect to independent variables such as x, y, . t. , are written as ux; ; uxx; ; uxy; ; uxxx ::: (1.0.1) or @ @2 @2 @3 u; u; u; u; : : : @x @x2 @y@x @x3 we write @2 u = (u ) = u; xy x y @y@x to indicate that the partial with respect to x is taken first. If we have a function of one variable, say x, then the only partial derivative of @f f(x), is just the derivative f 0(x) and equations involving functions of one variable @x and their derivatives are called ordinary differential equations (ODEs). Some example of PDEs are 2 ux + tutt = t (1.0.2a) 1 2 ut − k uxx = cos t (1.0.2b) 2 utt − c uxx + f (x; t) = 0 (1.0.2c) ut + uux + uxxx = 0 (1.0.2d) uxx (x; y) + uyy (x; y) = 0; (1.0.2e) 3 utt (x; t) = uxx (x; t) − u (x; t) ; (1.0.2f) 2 2 ut (x; t) + x + t ux (x; t) = 0; (1.0.2g) Definition 1.2. The order of an ODE, or a PDE equation is the maximal number of derivatives (or partial derivatives, respectively) taken with respect to the independent variable(s). Equation (1.0.2a), is of second order because u has been differentiated twice with respect to t; equation (1.0.2d) has three x as subscripts, indicating a PDE of third order, and in (1.0.2g), the first u has been differentiated just once, and so has the second u; this is a PD equation of first order. We now define what it means for a differential operator L to be linear. Definition 1.3. L is linear if for any fucntions u and v and constant c we have L(u + v) = Lu + Lv & L(cu) = cLu We show examples of linearity and non-linearity of the following PDEs in a slightly different way; we examine the factorization of the differential operator L and u in the following equations, ux + uy = 0 transport (1.0.3a) 2 ux − yuy = 0 transport (1.0.3b) ux + uuy = 0 shock wave (1.0.3c) We will write these equations in the form Lu = ux + uy . If we can factor out u completely from a differential operator L; that is, separate L from u with no u in L, then L is linear. We show this with equations (1.0.3a), (1.0.3b) and (1.0.3c) @ @ Lu = u + u = + u (1.0.4a) x y @x @y @ @ Lu = u − yu = − y u (1.0.4b) x y @x @y @ @ Lu = u + uu = + u u (1.0.4c) x y @x @y The differential operator in (1.0.4a) is linear because there is no u in it; that is, no u @ @ in the differential operator @x + @y ; in (1.0.4b), the y in the differential operator makes no difference, so ux − yuy is linear. And equation (1.0.4c) is not linear because there is a u in the differential operator after factoring. 2 3 4 p Here are some operations that give nonlinear operators: ux; u ; uxy :::, uxx; ln(u), sin(u), cos(u);:::, etc. Using the definition of linearity we show Lu = ux + uy is linear. For dependent functions v and u and c constant, we have L(u + v) = (u + v)x + (u + v)y = ux + vx + uy + vy = (ux + uy) + (vx + vy) = Lu + Lv; 3 and L(cu) = (cu)x + (cu)y = cux + cuy = c(ux + uy) = cL(u) this proves linearity of Lu = ux + uy. Consider this nonlinear differential expression Lu = ux + uy + 1, then for dependent functions u and v, L(u + v) = (u + v)x + (u + v)y + 1 = ux + vx + uy + vy + 1 but Lu + Lv = ux + uy + 1 + vx + vy + 1 = ux + vx + uy + vy + 2 this means L(u + v) 6= Lu + Lv; hence, the operator L is nonlinear. However, given ut − uxx + 1 = 0 we can move the constant 1 to the right side ut − uxx = −1; and think of it as Lu = −1; the left side is linear (u can be factored from the differential operator L, or we can use the the definition of linearity above), so we have a linear equation. Definition 1.4. Suppose L is a linear operator, then we define Lu = g, where g is a function containing only independent variables such as x; y; z; : : : (u such as in ux is not independent), then equation Lu = g is called an inhomogeneous equation if 4 g 6= 0 and Lu = g is homogeneous if g = 0. Examples of inhomogeneous (homogeneous) equations are 2 2 2 2 cos xy ux − y uy = tan x + y (1.0.5a) 2 utt − c uxx + f (x; t) = 0; where f (x; t) 6= 0 (1.0.5b) Both equations, (1.0.5a and (1.0.5b) are inhomogeneous, while uxx(x; y) + uyy(x; y) = 0 is a homogeneous equation. 5 CHAPTER 2 Reduction to Canonical Form Figure 1: Chain rule This