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{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "T imes" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 42 "Section \+ 3.3 Eigenvalues and Eigenfunctions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT 269 29 "Maple Packages in Section 3.3" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 402 "We have found that having an orthogonal family in a vector space is useful for the purposes of representing other eleme nts of the space. If one were looking for a basis for the space, an or thogonal family would be the ideal choice. We have one method for gene rating an orthogonal family: perform the Gramm-Schmidt process. In thi s worksheet we introduce another method for getting an orthogonal fami ly." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "W e compute eigenvectors for self-adjoint, linear transformations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 11 "Definiti on:" }{TEXT -1 35 " A function A on a vector space is " }{TEXT 257 6 " linear" }{TEXT -1 3 " if" }}{PARA 0 "" 0 "" {TEXT -1 10 " " } {XPPEDIT 18 0 "A(lambda*x+y) = lambda*A(x)+A(y);" "6#/-%\"AG6#,&*&%'la mbdaG\"\"\"%\"xGF*F*%\"yGF*,&*&F)F*-F%6#F+F*F*-F%6#F,F*" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "for all numbers " }{XPPEDIT 18 0 "lam bda;" "6#%'lambdaG" }{TEXT -1 17 " and all vectors " }{TEXT 273 1 "x" }{TEXT -1 5 " and " }{TEXT 274 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "Examples." }}{PARA 0 " " 0 "" {TEXT 379 1 "1" }{TEXT -1 72 ". Let A be a square matrix define d on a finite dimensional vector space." }}{PARA 0 "" 0 "" {TEXT 380 2 "2." }{TEXT -1 27 " Let the space be C ''([0, " }{XPPEDIT 18 0 "infi nity;" "6#%)infinityG" }{TEXT -1 10 " )) and A(" }{TEXT 270 1 "f" } {TEXT -1 4 ") = " }{TEXT 271 1 "f" }{TEXT -1 12 " '' for any " }{TEXT 275 1 "f" }{TEXT -1 14 " in the space." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 262 11 "Definition:" }{TEXT -1 79 " A linear function A is self-adjoint on the space \{ E, < *, * > \} if , for al l " }{TEXT 276 1 "x" }{TEXT -1 5 " and " }{TEXT 277 1 "y" }{TEXT -1 30 " in the domain of the operator" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }{XPPEDIT 18 0 " = ;" "6#/-%$<,>G6$%#AxG6#%\"yG-F %6$%\"xG6#%#AyG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 10 "Example 3." }}{PARA 0 "" 0 "" {TEXT -1 205 "One class of self-adjoint linear functions consists of matrices w hich have real number entries and which are symmetric about the main d iagonal. To be specific, we illustrate with a two dimensional example. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "First , we choose all the entries to be real numbers, not complex numbers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "assume(alpha,real):\nassum e(beta,real):\nassume(delta,real):\n" }}}{PARA 0 "" 0 "" {TEXT -1 34 " Now, we define a symmetric matrix." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=Matrix([[alpha,beta],[beta,delta]]);" }}}{PARA 0 " " 0 "" {TEXT -1 26 "Take two vectors, X and U." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "X:=;\nU:=;" }}}{PARA 0 "" 0 "" {TEXT -1 70 "Compute the dot product of AX with U and the dot product of X w ith AU." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "AXU:=DotProduct(A .X,U);\nXAU:=DotProduct(X,A.U);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "Show \+ that these two are equal." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify( AXU-XAU );" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 374 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 166 "As a second example, we take A to be a differential operator on a subset of C ''( [0, 1]) with the usual dot product. The subset of C ''([0, 1]) consist s of functions " }{TEXT 278 1 "f" }{TEXT -1 11 " such that " }{TEXT 279 1 "f" }{TEXT -1 6 "(0) = " }{TEXT 280 1 "f" }{TEXT -1 34 "(1) = 0. The definition of A is A(" }{TEXT 281 1 "f" }{TEXT -1 4 ") = " } {TEXT 282 1 "f" }{TEXT -1 83 " ''. We establish that this transformati on is self adjoint. That is, we show that " }}{PARA 0 "" 0 "" {TEXT -1 36 "=, or " }{XPPEDIT 18 0 "Int(diff( f(x),`$`(x,2))*y(x),x = 0 .. 1) = Int(f(x)*diff(y(x),`$`(x,2)),x = 0 . . 1);" "6#/-%$IntG6$*&-%%diffG6$-%\"fG6#%\"xG-%\"$G6$F.\"\"#\"\"\"-%\" yG6#%\"xGF3/F7;\"\"!F3-F%6$*&-%\"fG6#F7F3-%%diffG6$-%\"yG6#%\"xG-%\"$G 6$FGF2F3/F7;F:F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 74 "To se e how this is done the reader needs to remember integration-by-parts: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }{XPPEDIT 18 0 "int(diff(u(x),x)*v(x),x = a .. b) = u(b)*v(b)-u (a)*v(a)-int(u*diff(v(x),x),x = a .. b);" "6#/-%$intG6$*&-%%diffG6$-% \"uG6#%\"xGF.\"\"\"-%\"vG6#F.F//F.;%\"aG%\"bG,(*&-F,6#F6F/-F16#F6F/F/* &-F,6#F5F/-F16#F5F/!\"\"-F%6$*&F,F/-F)6$-F16#F.F.F//F.;F5F6FB" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We must consider" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 12 " " }{XPPEDIT 18 0 "int(diff(f(x),`$`(x,2))*g (x),x = 0 .. 1)-int(f(x)*diff(g(x),`$`(x,2)),x = 0 .. 1);" "6#,&-%$int G6$*&-%%diffG6$-%\"fG6#%\"xG-%\"$G6$F.\"\"#\"\"\"-%\"gG6#F.F3/F.;\"\"! F3F3-F%6$*&-F,6#F.F3-F)6$-F56#F.-F06$F.F2F3/F.;F9F3!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Usi ng the integration by parts with " }{TEXT 283 1 "u" }{TEXT -1 3 " = " }{TEXT 284 1 "f" }{TEXT -1 8 " '' and " }{TEXT 285 1 "v" }{TEXT -1 3 " = " }{TEXT 286 1 "g" }{TEXT -1 23 ", the first integral is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " } {TEXT 287 1 "f" }{TEXT -1 6 " '(1) " }{TEXT 288 1 "g" }{TEXT -1 6 "(1) - " }{TEXT 289 1 "f" }{TEXT -1 6 " '(0) " }{TEXT 290 1 "g" }{TEXT -1 7 "(0) - " }{XPPEDIT 18 0 "int(diff(f,x)*diff(g,x),x = a .. b);" "6#- %$intG6$*&-%%diffG6$%\"fG%\"xG\"\"\"-F(6$%\"gGF+F,/F+;%\"aG%\"bG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 51 "On the other hand, usin g integration by parts with " }{TEXT 291 1 "u" }{TEXT -1 3 " = " } {TEXT 292 1 "g" }{TEXT -1 8 " '' and " }{TEXT 293 1 "v" }{TEXT -1 3 " \+ = " }{TEXT 294 1 "f" }{TEXT -1 36 " we have that the second integral i s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }{TEXT 295 1 "g" }{TEXT -1 6 " '(1) " }{TEXT 296 1 "f" }{TEXT -1 6 "(1) - " }{TEXT 297 1 "g" }{TEXT -1 6 " '(0) " }{TEXT 298 1 "f" } {TEXT -1 6 "(0) - " }{XPPEDIT 18 0 "int(diff(f,x)*diff(g,x),x = 0 .. 1 );" "6#-%$intG6$*&-%%diffG6$%\"fG%\"xG\"\"\"-F(6$%\"gGF+F,/F+;\"\"!F, " }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 60 "When we subtract the second integral from the first, we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 299 1 "f" } {TEXT -1 6 " '(1) " }{TEXT 300 1 "g" }{TEXT -1 6 "(1) - " }{TEXT 301 1 "f" }{TEXT -1 6 " '(0) " }{TEXT 302 1 "g" }{TEXT -1 6 "(0) - " } {TEXT 303 1 "g" }{TEXT -1 6 " '(1) " }{TEXT 304 1 "f" }{TEXT -1 6 "(1) + " }{TEXT 305 1 "g" }{TEXT -1 11 " '(0) f(0)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "This last line will be ze ro if " }{TEXT 306 1 "f" }{TEXT -1 5 " and " }{TEXT 307 1 "g" }{TEXT -1 71 " satisfy the boundary conditions to be in the domain of A, that is, if " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 308 1 "f" }{TEXT -1 10 "(1) = 0 = " }{TEXT 309 1 "f" }{TEXT -1 8 "(0) and " }{TEXT 310 1 "g" }{TEXT -1 10 "(1) = 0 = " }{TEXT 311 1 "g" }{TEXT -1 4 "(0)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 266 "We have \+ established that this A -- take two derivatives, with boundary conditi ons -- is self adjoint. (The reader might observe in passing that ther e are other boundary conditions that would achieve this same condition . We will see some of these in future problems.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 11 "Definition:" }{TEXT -1 12 " The number " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 33 " is an eigenvalue and the vector " }{TEXT 313 1 "v" }{TEXT -1 52 " is an eigenvector of the linear transformation A if" }}{PARA 0 "" 0 "" {TEXT -1 12 " A(" }{TEXT 314 1 "v" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 315 1 "v " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 71 "The numbers -1 \+ and -2 are eigenvalues corresponding to the eigenvectors" }}{PARA 0 " " 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "matrix([[1], [1]]);" " 6#-%'matrixG6#7$7#\"\"\"7#F(" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "mat rix([[1], [-1]]);" "6#-%'matrixG6#7$7#\"\"\"7#,$F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "for the matrix A given as" }}{PARA 0 "" 0 "" {TEXT -1 35 " A = " }{XPPEDIT 18 0 "matrix([[-3/2, 1/2], [1/2, -3/2]]);" "6#-%'matrixG6#7$7$,$*&\"\" $\"\"\"\"\"#!\"\"F-*&F+F+F,F-7$*&F+F+F,F-,$*&F*F+F,F-F-" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "We us e Maple to give those eigenvalues and eigenvectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=Matrix([[ -3/2,1/2],[1/2,-3/2]]);" }}}{PARA 0 "" 0 "" {TEXT -1 277 "Maple produc es a list. The first entry is interpreted as follows: -1 is an eigenva lue of multiplicity 1 and corresponding to the eigenvector [1, 1]. The second entry is interpreted asfollows: -2 is an eigenvalue of multipl icity 1 and corresponding to the eigenvector [-1, 1]. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Eigenvectors(A, output='list');" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 375 9 "Example 6" }} {PARA 0 "" 0 "" {TEXT -1 20 "Each of the numbers " }{XPPEDIT 18 0 "-pi ^2;" "6#,$*$%#piG\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-4*Pi^2; " "6#,$*&\"\"%\"\"\"*$%#PiG\"\"#F&!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "-9*Pi^2;" "6#,$*&\"\"*\"\"\"*$%#PiG\"\"#F&!\"\"" }{TEXT -1 7 ", \+ ..., " }{XPPEDIT 18 0 "-n^2*Pi^2;" "6#,$*&%\"nG\"\"#%#PiGF&!\"\"" } {TEXT -1 58 " ... is an eigenvalue corresponding to the eigenfunction " }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "sin(n*Pi *x);" "6#-%$sinG6#*(%\"nG\"\"\"%#PiGF(%\"xGF(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "for the linear operator of Example 4 above. We \+ verify this here." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "assume( n,integer):\nu:=x->sin(n*Pi*x);\nlambda:=-n^2*Pi^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "is(diff(u(x),x,x)=lambda*u(x));\nis(u(0)= 0);\nis(u(1)=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 6 "Remark" } {TEXT -1 93 ": We have presented the eigenvalues and eigenvectors for \+ these two linear transformations as " }{TEXT 258 14 "faits accompli" } {TEXT -1 271 ". The process for getting eigenvalues and eigenvectors f or a matrix is the subject of another course. The process of getting t he eigenvalues and eigenfunctions for this differential operator is a \+ subject for this course, and we illustrate how to derive the results h ere. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT 263 9 "Question:" }{TEXT 318 1 " " }{TEXT -1 72 "What are all the eigenvalues of the selfadjoint, Sturm-Liouvill e Problem" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 319 1 "y " }{TEXT -1 6 " '' = " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 1 " " }{TEXT 320 1 "y" }{TEXT -1 7 ", with " }{TEXT 321 1 "y" }{TEXT -1 6 "( 0) = " }{TEXT 322 1 "y" }{TEXT -1 8 "(1) = 0?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We break the problem into two cases. First, suppose that " }{XPPEDIT 18 0 "mu;" "6#%#muG" } {TEXT -1 41 " is positive. To remind us of this, take " }{XPPEDIT 18 0 "mu = lambda^2;" "6#/%#muG*$%'lambdaG\"\"#" }{TEXT -1 25 " . Thus, w e seek numbers " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 10 " such that" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 323 1 "y " }{TEXT -1 6 " '' = " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"# " }{TEXT -1 1 " " }{TEXT 324 1 "y" }{TEXT -1 6 " with " }{TEXT 325 1 " y" }{TEXT -1 6 "(0) = " }{TEXT 326 1 "y" }{TEXT -1 8 "(1) = 0." }} {PARA 0 "" 0 "" {TEXT -1 126 "The differential equation is a second or der constant coefficient equation. We know how to solve it. Solutions \+ are of the form " }{XPPEDIT 18 0 "exp(r*x)" "6#-%$expG6#*&%\"rG\"\"\"% \"xGF(" }{TEXT -1 51 ". In this case, the general solution is of the f orm" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{TEXT 328 1 "y" }{TEXT -1 1 "(" }{TEXT 377 1 "x" }{TEXT -1 11 ") = A exp( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 329 2 " x" }{TEXT -1 11 ") + B exp ( " }{XPPEDIT 18 0 "-lambda;" "6#,$%'lambdaG!\"\"" }{TEXT -1 1 " " } {TEXT 330 1 "x" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 12 "To ask that " }{TEXT 331 1 "y" }{TEXT -1 41 "(0) = 0 asks that 0 = A + B. To ask that " }{TEXT 332 1 "y" }{TEXT -1 28 "(1) = 0 asks that 0 = A exp (" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 10 ") + B/exp(" } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 100 ") . The only solu tion is A = 0 = B. This is the trivial solutions and will not be of in terest to us." }}{PARA 0 "" 0 "" {TEXT -1 29 " The second case is \+ that " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 41 " is negative. To r emind us of this, take " }{XPPEDIT 18 0 "mu = -lambda^2;" "6#/%#muG,$* $%'lambdaG\"\"#!\"\"" }{TEXT -1 24 ". Thus, we seek numbers " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 11 " such that" }} {PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 333 1 "y" }{TEXT -1 6 " '' = " }{XPPEDIT 18 0 "-lambda^2;" "6#,$*$%'lambdaG\"\"#!\"\"" } {TEXT -1 1 " " }{TEXT 334 1 "y" }{TEXT -1 6 " with " }{TEXT 335 1 "y" }{TEXT -1 6 "(0) = " }{TEXT 336 1 "y" }{TEXT -1 8 "(1) = 0." }}{PARA 0 "" 0 "" {TEXT -1 79 "We also know how to solve this differential equ ation. Solutions are of the form" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }{TEXT 338 1 "y" }{TEXT -1 1 "(" }{TEXT 376 1 "x" }{TEXT -1 10 ") = A sin(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 337 1 "x" }{TEXT -1 10 ") + B cos(" }{XPPEDIT 18 0 "lambda; " "6#%'lambdaG" }{TEXT -1 1 " " }{TEXT 339 1 "x" }{TEXT -1 2 ")." }} {PARA 0 "" 0 "" {TEXT -1 12 "To ask that " }{TEXT 340 1 "y" }{TEXT -1 42 "(0) = 0 asks that 0 = B. To also ask that " }{TEXT 341 1 "y" } {TEXT -1 28 "(1) = 0 asks that 0 = A sin(" }{XPPEDIT 18 0 "lambda;" "6 #%'lambdaG" }{TEXT -1 82 "). We don't want to have the trivial solutio ns again, so it must be that 0 = sin(" }{XPPEDIT 18 0 "lambda;" "6#%' lambdaG" }{TEXT -1 81 ") . But we know all the places where the sine \+ function is zero. It must be that " }{XPPEDIT 18 0 "lambda = n*Pi;" "6 #/%'lambdaG*&%\"nG\"\"\"%#PiGF'" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 " mu = -n^2*Pi^2;" "6#/%#muG,$*&%\"nG\"\"#%#PiGF(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 55 "The answer to the question is that the \+ eigenvalues are " }{XPPEDIT 18 0 "-n^2*Pi^2;" "6#,$*&%\"nG\"\"#%#PiGF& !\"\"" }{TEXT -1 32 " and the eigenfuntions are sin(" }{XPPEDIT 18 0 "n*Pi*x;" "6#*(%\"nG\"\"\"%#PiGF%%\"xGF%" }{TEXT -1 2 ")." }}{PARA 0 " " 0 "" {TEXT -1 29 "Somehow, this is no surprise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "We do not forget the purp ose of this development." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 260 7 "Fact 1:" }{TEXT -1 98 " Eigenvalues corresponding \+ to self-adjoint transformations are real numbers - not complex numbers ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 "Fact 2:" }{TEXT -1 101 " Eigenvectors for self adjoint transformations cor responding to different eigenvalues are orthogonal." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "Revisit Examples 3 and 4. Note that the eigenvectors are orthogonal. In the matrix example, \+ this is seen quickly. For the differential operator, recall that the d ot product is an integral." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "assume(m,integer):\n\004int(sin(n*Pi*x)*sin(m*Pi*x),x=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 378 6 "Remark" }{TEXT -1 210 ". Concer ning Fact 2 above, suppose we have two eigenvectors corresponding to o ne eigenvalue. The eigenvectors found may not be orthogonal. Not to wo rry! Perform the Gramm-Schmidt process to these eigenvectors. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "Most of \+ the partial differential equations that arise in this set of lecture n otes will generate a special type of differential equation that is cal led a Sturm-Liouville problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 48 "Definition of a regular Sturm-Liouville \+ problem:" }{TEXT -1 14 " Suppose that " }{TEXT 342 1 "s" }{TEXT -1 2 " , " }{TEXT 343 1 "s" }{TEXT -1 8 " ', and " }{TEXT 344 1 "q" }{TEXT -1 20 " are continuous and " }{XPPEDIT 18 0 "s(x) <> 0;" "6#0-%\"sG6#% \"xG\"\"!" }{TEXT -1 5 " for " }{TEXT 345 1 "x" }{TEXT -1 88 " in [a,b ]. In the presence of appropriate boundary conditions, the differentia l operator" }}{PARA 0 "" 0 "" {TEXT -1 13 " L(f) = (" }{TEXT 346 1 "s" }{TEXT -1 1 " " }{TEXT 347 1 "f" }{TEXT -1 8 " ') ' - " }{TEXT 348 1 "q" }{TEXT -1 1 " " }{TEXT 349 1 "f" }}{PARA 0 "" 0 "" {TEXT -1 94 "is self adjoint in C ''([a,b]). Example of appropriate boundary co nditions are that functions " }{TEXT 350 1 "f" }{TEXT -1 38 " in the d omain of L should satisfy are" }}{PARA 0 "" 0 "" {TEXT -1 4 "(1) " } {TEXT 351 1 "f" }{TEXT -1 5 "(0) =" }{TEXT 352 2 " f" }{TEXT -1 11 "(1 ) = 0, or" }}{PARA 0 "" 0 "" {TEXT -1 4 "(2) " }{TEXT 353 1 "f" } {TEXT -1 8 " '(0) = " }{TEXT 354 1 "f" }{TEXT -1 13 " '(1) = 0, or" }} {PARA 0 "" 0 "" {TEXT -1 4 "(3) " }{TEXT 355 1 "f" }{TEXT -1 6 "(0) = \+ " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 " " }{TEXT 356 1 "f " }{TEXT -1 10 " '(0) and " }{TEXT 357 1 "f" }{TEXT -1 6 "(1) = " } {XPPEDIT 18 0 "-alpha;" "6#,$%&alphaG!\"\"" }{TEXT -1 1 " " }{TEXT 358 1 "f" }{TEXT -1 9 " '(1), or" }}{PARA 0 "" 0 "" {TEXT -1 7 "(4) if " }{TEXT 359 1 "s" }{TEXT -1 6 "(0) = " }{TEXT 360 1 "s" }{TEXT -1 10 "(1), then " }{TEXT 361 1 "f" }{TEXT -1 6 "(0) = " }{TEXT 362 1 "f " }{TEXT -1 8 "(1) and " }{TEXT 363 1 "f" }{TEXT -1 8 " '(0) = " } {TEXT 364 1 "f" }{TEXT -1 60 " '(1). (These last are called periodic b oundary conditions.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 365 9 "Example 7" }}{PARA 0 "" 0 "" {TEXT -1 95 "We provide tw o eigenvectors (eigenfunctions) for each eigenvalue of the Sturm-Liouv ille problem" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{TEXT 366 1 "f" }{TEXT -1 6 " '' = " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" } {TEXT -1 1 " " }{TEXT 367 1 "f" }{TEXT -1 5 ", " }{TEXT 368 1 "f" } {TEXT -1 7 "(-1) = " }{TEXT 369 1 "f" }{TEXT -1 6 "(1), " }{TEXT 370 1 "f" }{TEXT -1 9 " '(-1) = " }{TEXT 371 1 "f" }{TEXT -1 6 " '(1)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "In this \+ example, there are an infinite number of eigenvalues, and for each one , there are two eigenfunctions. We will see that the eigenvalues are \+ " }{XPPEDIT 18 0 "-n^2*pi^2;" "6#,$*&%\"nG\"\"#%#piGF&!\"\"" }{TEXT -1 22 " , for each integer n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(n,integer);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "y1:=x->sin(n*Pi*x);\ny2:=x->cos(n*Pi*x);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "A:=y->diff(y(x),x$2)+n^2*Pi^2*y(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A(y1);\nA(y2);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "y1(-1); y1(1);\nD(y1)(-1);D(y1)(1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "y2(-1); y2(1);\nD(y2)(-1);D( y2)(1);" }}}{EXCHG }{EXCHG }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 93 "For later reference, we give the eigenvalues and e igenfunctions for the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 18 " " }{TEXT 372 1 "y" }{TEXT -1 6 " '' = " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 373 2 " y" }}{PARA 0 "" 0 "" {TEXT -1 91 "for a variety of boundary conditions. Other boundary conditions not mentioned are possible." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 264 30 "Eigenvalues and Eigenfunctions" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 265 34 "Dirichlet Zero Boundary Conditions" }}{PARA 0 "" 0 "" {TEXT -1 35 "Bo undary condition: y(0) = y(L) = 0" }}{PARA 0 "" 0 "" {TEXT -1 14 "Eige nvalues: n" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 17 "/L, n = 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 21 "Eigenfunctions: sin(n" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 20 "x/L) , n = 1, 2, ..." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 37 "Neumann Insulated Boundary Conditions" } }{PARA 0 "" 0 "" {TEXT -1 37 "Boundary condition: y'(0) = y'(L) = 0" } }{PARA 0 "" 0 "" {TEXT -1 14 "Eigenvalues: n" }{XPPEDIT 18 0 "Pi;" "6# %#PiG" }{TEXT -1 20 "/L, n = 0, 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 21 "Eigenfunctions: cos(n" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 23 "x/L) , n = 0, 1, 2, ..." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 28 " Periodic Boundary Conditions" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Boundary condition: y(0) = y(L), y'(0) = y'(L)" }}{PARA 0 "" 0 "" {TEXT -1 14 "Eigenvalues: n" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 19 "/L, n = 0,1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 21 "Eigenfuncti ons: sin(n" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 29 "/L) , n = 1, \+ 2, ... and cos(n" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 22 "/L) , n = 0, 1, 2, ..." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 268 25 "Mixed Bounda ry Conditions" }}{PARA 0 "" 0 "" {TEXT -1 37 "Boundary condition: y(0) = y'(L) = 0," }}{PARA 0 "" 0 "" {TEXT -1 14 "Eigenvalues: (" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 7 "/2 + n " }{XPPEDIT 18 0 "Pi ;" "6#%#PiG" }{TEXT -1 21 ")/L, n = 0, 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 21 "Eigenfunctions: sin((" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 7 "/2 + n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 24 ") x \+ /L, n = 0, 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 2 "OR" }}{PARA 0 "" 0 "" {TEXT -1 37 "Boundary condition: y'(0) = y(L) = 0," }}{PARA 0 "" 0 "" {TEXT -1 14 "Eigenvalues: (" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 7 "/2 + n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 21 ")/L, n = 0, 1, 2, ..." }}{PARA 0 "" 0 "" {TEXT -1 21 "Eigenfunctions: cos( (" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 7 "/2 + n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 24 ") x /L, n = 0, 1, 2, ..." }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "EMAIL: herod@math. gatech.edu or jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: \+ http://www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 36 "Copyright \251 2003 by James V. Herod " }}{PARA 258 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 36 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }