# nonhomogeneous heat equation separation of variables

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298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 One of the classic PDE’s equations is the heat equation. 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 722 941 667 611 611 611 611 333 333 333 333 778 778 778 778 778 778 778 606 778 778 22 0 obj endobj 833 611 556 833 833 389 389 778 611 1000 833 833 611 833 722 611 667 778 778 1000 >> endobj 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 667 667 333 606 333 606 500 278 444 463 407 500 389 278 500 500 278 278 444 278 778 Heat ﬂow with sources and nonhomogeneous boundary conditions We consider ﬁrst the heat equation without sources and constant nonhomogeneous boundary conditions. 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 0 0 0 0 853 0 0 0 0 0 0 0 0 0 0 0 /Subtype/Type1 /Widths[250 0 0 376 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /BaseFont/FMLSVH+URWPalladioL-Roma /Subtype/Type1 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 xڽW[o�D~�W� G��{� @�V\$�ۉБ(n�6�\$�Ӵ���z���z@�%^gwg�����J���~�}���c3��h�1J��Q"(Q"Z��{��.=U�y�pEcEV�`4����sZ���/���ʱ8=���>+W��~Z�8�UE���I���@(�q��K�R�ȏ.�>��8Ó�N������+.p����"..�FZq�W����9?>�K���Ed� �:�x�����h.���K��+xwos��]�V� endstream /F7 29 0 R /FontDescriptor 40 0 R The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.. /FirstChar 32 26 0 obj /Encoding 7 0 R 1 Separation of Variables in Cylindrical Coordinates We consider two dimensional problems with cylindrical symmetry (no dependence on z).Our variables are s in the radial direction and φ in the azimuthal direction. >> Boundary Value Problems (using separation of variables). Section 4.6 PDEs, separation of variables, and the heat equation. 3 The method may work for both homogeneous (G = 0) and nonhomogeneous (G ̸= 0) PDE’s Chapter 12 PDEs in Rectangles 1 2-D Second Order Equations: Separation of Variables 1.A second order linear partial di erential equation in two variables xand yis A @2u @x 2 + B @ 2u @x@y + C @u @y + D @u @x + E @u @y + Fu= G: (1) 2.If G= 0 we say the problem is homogeneous otherwise it is nonhomogeneous. /FontDescriptor 24 0 R 7 0 obj /Type/Font 29 0 obj /Widths[333 611 611 167 333 611 333 333 333 0 333 606 0 667 500 333 333 0 0 0 0 0 Unfortunately, this method requires that both the PDE and the BCs be … Initial Value Problems Partial di erential equations generally have lots of solutions. /BaseFont/GUEACL+CMMI10 We consider a general di usive, second-order, self-adjoint linear IBVP of the form u /Filter /FlateDecode Nonhomogeneous Problems. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis The LaPlace equation in cylindricalcoordinates is: 1 s ∂ ∂s s ∂V(s,φ) ∂s + 1 s2 ∂2V(s,φ) ∂φ2 =0 We try to ﬁnd a solution of the form V (s,φ)=F(s)G(φ). >> Gn�U�����O7ٗ�P �M�[\4rd|M��NN2���R�Fp����!�v�v�r&p�!h�k���%@��D�Z�\$l�CG�uP�X�֞��s� w�-p^�r� �Ǣ�Z��N��ߪem�w��Ø%��9���X�?��c�Hbp��}��0����f��{ tZz}����J���T��&:%`�s.�xNv�\$�6��#�\$/���6��F�첛�dμ��!��P��vQ0]%�9�{�ܯ:n�|���U^��6M|}VB��*O�����������6�q��I92���+zQZ��}��CG��U��M\$�:��IB0�Ph�������n�v��M�� ;�sIo���#`Ҧ=0fS��!뗽7n�U:!�u,g�\$ܼ�q��wpl�6;��66L� �BU�cF�R��7����Ҏ��tS̋�e��LJ"��C�����ޚK����H�#�}�ɲS>��r{=��RH�N����eJ��SĐ�24�e宸��@����%k�"��3��l��D����? 0 0 0 0 0 0 0 333 208 250 278 371 500 500 840 778 278 333 333 389 606 250 333 250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 159/Ydieresis 161/exclamdown/cent/sterling/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis] /F3 16 0 R /Subtype/Type1 >> 13 0 obj 277.8 500] Note: 2 lectures, §9.5 in , §10.5 in . So, we’re going to need to deal with the boundary conditions in some way before we actually try and solve this. /Widths[250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 285 0 0 0 << 500 500 1000 500 500 333 1000 611 389 1000 0 0 0 0 0 0 500 500 606 500 1000 333 998 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] 278 444 556 444 444 444 444 444 606 444 556 556 556 556 500 500 500] In the previous chapter we saw that when solving a wave or heat equation it may be necessary to first ... sides and a nonhomogeneous Dirichlet boundary condition on the fourth side. 296 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 500 747 722 611 667 and consequently the heat equation (2,3,1) implies that 2.3.2 Separation ofVariables where ¢(x) is only a function of x and G(t) only a function of t, Equation (2,3.4) must satisfy the linear homogeneous partial differential equation (2.3,1) and bound­ ary conditions (2,3,2), but for themoment we set aside (ignore) the initial condition, 400 606 300 300 333 603 628 250 333 300 333 500 750 750 750 444 778 778 778 778 778 << 778 778 778 667 604 556 500 500 500 500 500 500 758 444 479 479 479 479 287 287 287 /Name/F7 x��ZKs���WpIOLo��.�&���2��I��L[�Ȓ*J�M}� �a�N���ƒ���w����FWO���{����HEjEu�X1�ڶjF�Tw_�Xӛ�����;1v!�MUض�m���������i��w���w��v������_7���~ս_�������`�K\�#�V��q~���N�I[��fs�̢�'X���a�g�k�4��Z�9 E�����ǰ�ke?Y}_�=�7����؅m߯��=. /BaseFont/WETBDS+URWPalladioL-Bold /Type/Font /Type/Font 2 Heat Equation 2.1 Derivation Ref: Strauss, Section 1.3. Lecture 21 Phys 3750 D M Riffe -1- 3/18/2013 Separation of Variables in Cylindrical Coordinates Overview and Motivation: Today we look at separable solutions to the wave equation in cylindrical coordinates. /FontDescriptor 15 0 R 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 endobj /Type/Font endobj 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 /BaseFont/IZHJXX+URWPalladioL-Ital /F5 22 0 R 2 For the PDE’s considered in this lecture, the method works. /Type/Font Chapter 5. /Name/F9 >> 424 331 827 0 0 667 0 278 500 500 500 500 606 500 333 747 333 500 606 333 747 333 >> B�0Нt���K�A������X�l��}���Q��u�ov��>��6η���e�6Pb;#�&@p�a♶se/'X�����`8?�'\{o�,��i�z? PDE & Complex Variables P4-1 Edited by: Shang-Da Yang Lesson 04 Nonhomogeneous PDEs and BCs Overview This lesson introduces two methods to solve PDEs with nonhomogeneous BCs or driving source, where separation of variables fails to deal with. /FirstChar 1 Three of the resulting ordinary differential equations are again harmonic-oscillator equations, but the fourth equation is our first /Encoding 7 0 R >> We only consider the case of the heat equation since the book treat the case of the wave equation. >> /LastChar 196 If \(u_1\) and \(u_2\) are solutions and \(c_1,c_2\) are constants, then \( u= c_1u_1+c_2u_2\) is also a solution. 5. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 endobj 255/dieresis] If we can solve (4), then the original non-homogeneous heat equation (1) can be easily recovered. 667 667 667 333 606 333 606 500 278 500 553 444 611 479 333 556 582 291 234 556 291 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 4.1 The heat equation Consider, for example, the heat equation ut = uxx, 0 < x < 1, t > 0 (4.1) Below we provide two derivations of the heat equation, ut ¡kuxx = 0 k > 0: (2.1) This equation is also known as the diﬀusion equation. 556 444 500 463 389 389 333 556 500 722 500 500 444 333 606 333 606 0 0 0 278 500 >> 9.3 Separation of variables for nonhomogeneous equations Section 5.4 and Section 6.5, An Introduction to Partial Diﬀerential Equa-tions, Pinchover and Rubinstein The method of separation of variables can be used to solve nonhomogeneous equations. 444 389 833 0 0 667 0 278 500 500 500 500 606 500 333 747 438 500 606 333 747 333 0 0 0 0 0 0 0 333 333 250 333 500 500 500 889 778 278 333 333 389 606 250 333 250 /BaseFont/GNMCTH+PazoMath-Italic 14/Zcaron/zcaron/caron/dotlessi/dotlessj/ff/ffi/ffl 30/grave/quotesingle/space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright/asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde stream 0 0 0 0 0 0 0 333 227 250 278 402 500 500 889 833 278 333 333 444 606 250 333 250 �E��H���4k_O��\$����>�P�i�죶����V��D�g ��l�z�Sj.���>�.���=�������O'01���:Λr,��N��K�^9����I;�&����r)#��|��^n�+����LfvX���mo�l>�q>�3�g����f7Gh=qJ������uD�&�����-���C,l��C��K�|��YV��߁x�iۮ�|��ES��͗���^�ax����i����� �4�S�]�sfH��e���}���oٔr��c�ұ���%�� !A� 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 A formal definition of dx as a differential (infinitesimal)is somewhat advanced. 0 0 0 0 666 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 747 0 0 0 0 0 0 0 0 0 0 0 0 0 0 881 0 So it remains to solve problem (4). 667 667 667 333 606 333 606 500 278 500 611 444 611 500 389 556 611 333 333 611 333 /LastChar 196 Homogeneous case. /LastChar 229 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 32 0 obj Example 1. /FontDescriptor 18 0 R /Subtype/Type1 Free ebook http://tinyurl.com/EngMathYT How to solve the heat equation by separation of variables and Fourier series. /FirstChar 32 287 546 582 546 546 546 546 546 606 556 603 603 603 603 556 601 556] 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 389 333 669 0 0 667 0 333 500 500 500 500 606 500 333 747 333 500 606 333 747 333 << >> 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 /Type/Font /FirstChar 1 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 /Font 36 0 R << The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as heat equation, wave equation, Laplace equation and Helmholtz equation. /Length 2096 400 606 300 300 333 556 500 250 333 300 333 500 750 750 750 500 722 722 722 722 722 Recall that separation of variables will only work if both the partial differential equation and the boundary conditions are linear and homogeneous. /Subtype/Type1 400 606 300 300 333 611 641 250 333 300 488 500 750 750 750 444 778 778 778 778 778 endobj /Filter[/FlateDecode] 25 0 obj The heat and wave equations in 2D and 3D 18.303 Linear Partial Diﬀerential Equations Matthew J. Hancock Fall 2006 1 2D and 3D Heat Equation Ref: Myint-U & Debnath §2.3 – 2.5 [Nov 2, 2006] Consider an arbitrary 3D subregion V of R3 (V ⊆ R3), with temperature u(x,t) deﬁned at … << 4.6.2 Separation of variables. ... We again try separation of variables and substitute a solution of the form . Partial differential equations. endobj 41 0 obj 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 Assume that the sides of the rod are insulated so that heat energy neither enters nor leaves the rod through its sides. endobj << /Differences[1/dotaccent/fi/fl/fraction/hungarumlaut/Lslash/lslash/ogonek/ring 11/breve/minus /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 Unformatted text preview: The Heat Equation Heat Flow and Diffusion Problems Purpose of the lesson: To show how parabolic PDEs are used to model heat‐flow and diffusion‐type problems. >> Since by translation we can always shift the problem to the interval (0, a) we will be studying the problem on this interval. >> %PDF-1.4 /BaseFont/OBFSVX+CMEX10 774 611 556 763 832 337 333 726 611 946 831 786 604 786 668 525 613 778 722 1000 >> /Type/Font Figure \(\PageIndex{1}\): A uniform bar of length \(L\) ... Our method of solving this problem is called separation of variables ... Nonhomogeneous Problems. /F4 19 0 R 10 0 obj 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 u(x,t) = X(x)T(t) etc.. 2) Find the ODE for each “variable”. /Subtype/Type1 0 0 688 0 586 618 0 0 547 0 778 0 0 0 880 778 0 702 0 667 416 881 724 750 0 0 0 0 "��X���V��'b�� /Name/F5 << Separation of Variables and Classical PDE’s Wave Equation Laplace’s Equation Summary Some Remarks 1 The method of separation of variables can only solve for some linear second order PDE’s, not all of them. /ProcSet[/PDF/Text/ImageC] /Name/F2 ��=�)@ o�'@PS��?N'�Ϙ5����%�2���2B���2�w�`o�E�@��_Gu:;ϞQ���\�v�zQ ���BIZ�����ǖ�����~���6���[��ëZ��Ҟb=�*a)������ �n�`9���a=�0h�hD��8�i��Ǯ i�{;Mmŏ@���|�Vj��7n�S+�h��. >> /BaseFont/RZEVDH+PazoMath /FontDescriptor 9 0 R /Encoding 26 0 R /Subtype/Type1 /FontDescriptor 28 0 R 883 582 546 601 560 395 424 326 603 565 834 516 556 500 333 606 333 606 0 0 0 278 These conditions are usually motivated by the physics and come in two varieties: initial conditions and boundary conditions. /BaseFont/UBQMHA+CMR10 /FontDescriptor 21 0 R 889 611 556 611 611 389 444 333 611 556 833 500 556 500 310 606 310 606 0 0 0 333 /F1 10 0 R 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] u(x, t) = ¢(x)G(I), (2.3.4) where ¢(x) is only a function of x and G(I) only a function of t. Equation (2.3.4) must satisfy the linear homogeneous partial differential equation (2.3.1) and bound­ endobj endobj 16 0 obj /LastChar 255 In the method of separation of variables, we attempt to determine solutions in the product form . << 333 333 556 611 556 556 556 556 556 606 556 611 611 611 611 556 611 556] 2.1.1 Diﬀusion Consider a liquid in which a dye is being diﬀused through the liquid. /Subtype/Type1 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 Seven steps of the approach of separation of Variables: 1) Separate the variables: (by writing e.g. The –rst problem (3a) can be solved by the method of separation of variables developed in section 4.1. Nonhomogeneous Equations and Variation of Parameters June 17, 2016 1 Nonhomogeneous Equations 1.1 Review of First Order Equations If we look at a rst order homogeneous constant coe cient ordinary di erential equation by0+ cy= 0: then the corresponding auxiliary equation 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 %PDF-1.2 Solving PDEs will be our main application of Fourier series. stream Thus the principle of superposition still applies for the heat equation (without side conditions). /FontDescriptor 12 0 R To specify a unique one, we’ll need some additional conditions. /LastChar 196 To introduce the idea of an Initial boundary value problem (IBVP). /Type/Encoding 3 0 obj << 296 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 444 747 778 667 722 /Encoding 7 0 R << 3) Determine homogenous boundary values to stet up a Sturm- Liouville << 778 778 778 667 611 500 444 444 444 444 444 444 638 407 389 389 389 389 278 278 278 /F8 32 0 R /LastChar 226 When the problem is not homogeneous due to a nonhomogeneous energy equation or boundary condition, the solution of a nonhomogeneous problem can be obtained by superposition … Consider the one-dimensional heat equation.The equation is 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 Solving Nonhomogeneous PDEs Separation of variables can only be applied directly to homogeneous PDE. 521 744 744 444 650 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 obj 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 778 611 556 722 778 333 333 667 556 944 778 778 611 778 667 556 611 778 722 944 722 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 x��XKo�F���Q�B�!�]�=��F��z�s�3��������3Үd����Gz�FEr��H�ˣɋ}�+T�9]]V Z����2jzs��>Z�]}&��S��� �� ��O���j�k�o ���7a,S Q���@U_�*�u-�ʫ�|�`Ɵfr҇;~�ef�~��� �淯����Иi�O��{w��žV�1�M[�R�X5QIL���)�=J�AW*������;���x! https://tutorial.math.lamar.edu/.../SolvingHeatEquation.aspx 0 0 0 528 542 602 458 466 589 611 521 263 589 483 605 583 500 0 678 444 500 563 524 "���������{�h��T4ݯw|I���r�|eRK��pN�ܦ"����-k[5��W�j�I�y+?�Y;"D"̿�w�ވƠ�+����H�F���0����΄v�C��4�l��Bڡ_�C��E�����Ub�wK�Y�ӎ��\ �����ne� �_�^-r�E��ʂ;#zi-�i�MF�ꈓ�SvN��@��>a6��ݭ�s��~�(���!+����KKg*/�g*+]R@��SnZ['����X)U��W9h�\$�MA �3�����yi�m_�%�(ɱ��}�L_�x�Ď��w��\������o�{:�#�G���*��R~(d��Jю��8VV�O��Ik(hE~#h�!E�Ѧ���� U�ߢk�4������<=�E!�{:o8mOF�Zғ�Z�C�Oy��NZI#}_�����HP��d�i�2],1Q�o��/�I�}9�x��`�2�L�5ۑ����ql'��\+�+T����t�u��ƴ\$��H�E��q������1*+@�\l�굨���ȵ八���Zq�M\��H��3��4�?���7(�#�D\$E�r�%Ev3���Ź@>D=>:wn&���e���_�6�y� �ߕX�9�}3�����L^M�d�J+����PK��������w�:���̈́ << 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 0 278] /Length 1243 << /LastChar 255 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 However, it can be generalized to nonhomogeneous PDE with homogeneous boundary conditions by solving nonhomo-geneous ODE in time. /BaseFont/BUIZMR+CMSY10 dx (and dy) can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. /FirstChar 1 /Subtype/Type1 Chapter 12 Solving Nonhomogeneous PDEs (Eigenfunction Expansions) 12.1 Goal We know how to solve di⁄usion problems for which both the PDE and the BCs are homogeneous using the separation of variables method. 778 778 778 778 667 611 611 500 500 500 500 500 500 778 444 500 500 500 500 333 333 Title: Solution of the Heat Equation with Nonhomogeneous BCs Author: MAT 418/518 Fall 2020, by Dr. R. L. Herman Created Date: 20200909134351Z << 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 606 500 500 500 500 500 500 500 500 500 500 250 250 606 606 606 444 747 778 611 709 /Widths[250 605 608 167 380 611 291 313 333 0 333 606 0 667 500 333 287 0 0 0 0 0 Separation of Variables . Separation of Variables At this point we are ready to now resume our work on solving the three main equations: the heat equation, Laplace’s equation and the wave equa-tion using the method of separation of variables. 42 0 obj R.Rand Lecture Notes on PDE’s 2 Contents 1 Three Problems 3 2 The Laplacian ∇2 in three coordinate systems 4 3 Solution to Problem “A” by Separation of Variables 5 4 Solving Problem “B” by Separation of Variables 7 5 Euler’s Diﬀerential Equation 8 6 Power Series Solutions 9 1 1D heat and wave equations on a ﬁnite interval In this section we consider a general method of separation of variables and its applications to solving heat equation and wave equation on a ﬁnite interval (a 1, a2). /Filter[/FlateDecode] (∗) Transformation of Nonhomogeneous BCs (SJF 6) Problem: heat flow in a rod with two ends kept at constant nonzero … /Name/F8 /Type/Encoding 791.7 777.8] 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 500 500 1000 500 500 333 1144 525 331 998 0 0 0 0 0 0 500 500 606 500 1000 333 979 /Name/F6 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 >> /FirstChar 33 Solving non-homogeneous heat equation with homogeneous initial and boundary conditions. 6 0 obj The transient one-dimensional conduction problems that we discussed so far are limited to the case that the problem is homogeneous and the method of separation of variables works. The basic premise is conservation of energy. 0 676 0 549 556 0 0 0 0 778 0 0 0 832 786 0 667 0 667 0 831 660 753 0 0 0 0 0 0 0 << 19 0 obj 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 /Type/Font /Length 1369 /Differences[33/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 778 944 709 611 611 611 611 337 337 337 337 774 831 786 786 786 786 786 606 833 778 where \(a\) is a positive constant determined by the thermal properties. 500 1000 500 500 333 1000 556 333 1028 0 0 0 0 0 0 500 500 500 500 1000 333 1000 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 /FirstChar 33 /Widths[333 528 545 167 333 556 278 333 333 0 333 606 0 667 444 333 278 0 0 0 0 0 >> /Name/F1 36 0 obj 130/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE /FirstChar 33 /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 Let us recall that a partial differential equation or PDE is an equation containing the partial derivatives with respect to several independent variables. 0 0 0 0 0 0 0 0 0 0 0 234 0 881 767] /Name/F3 778 1000 722 611 611 611 611 389 389 389 389 833 833 833 833 833 833 833 606 833 /Type/Font endobj This is the heat equation. endobj /LastChar 196 The heat equation is linear as \(u\) and its derivatives do not appear to any powers or in any functions. 147/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark/scaron/guilsinglright/oe /FirstChar 33 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 endobj /F2 13 0 R 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 Separation of Variables and Heat Equation IVPs 1. stream /LastChar 255 0 0 0 0 0 0 0 0 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 487 0 0 0 0 0 0 0 0 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 Solution of the HeatEquation by Separation of Variables The Problem Let u(x,t) denote the temperature at position x and time t in a long, thin rod of length ℓ that runs from x = 0 to x = ℓ. endobj /Name/F4 << /FontDescriptor 31 0 R Suppose a differential equation can be written in the form which we can write more simply by letting y = f(x): As long as h(y) ≠ 0, we can rearrange terms to obtain: so that the two variables x and y have been separated. 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 Diﬀusion consider a liquid in which a dye is being diﬀused through the.. 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