Introduction to Partial Differential Equations
Introduction to Partial Differential Equations MATH 4163
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PDE Math 4163001 Fall 2011 Section 2 Classi cation of Second Order PDES A second order linear PDE for an unknown u uac7 y has the form aumbuxycuyyduweuyfug 21 so where a7 b7 07 cl7 67 f7 9 are constants or functions of 7 y7 but do not depend on u and or its derivatives We use the following terminology l The a um7 b um and c uyy are called the highest or second order terms 2 The other terms are called terms of lower order 3 g is called the inhomogeneous term and we call the equation homoge neous if g 07 otherwise the equation is called inhomogeneous We want to nd a characterization in terms of the coef cients of the second order derivatives a7 b and c which is independent of a change of variables Let E 06 531 y w 6y with 06 det fy 6 y 0 Now with 7y ozac y nvw5y we get um oz U5 7 Un uy U5 6 Un and so um agU 20WU n 72Unn my aU 605 Y j n 67mm 1 uyy QUgg 256U n 62Unn Inserting these derivatives into the PDE gives aumbuxycuyy a a2 b oz 052U55 2a 047 b60z 57 2056U5n CWQ I967 062U7m ELUgg EUgn Unn With some algebra we nd 132 4amp6 b2 4acoz6 7mg b2 4m det E So we nd that under all possible transformation the sign of D b2 4cm is invariant under a change of variables We call the equation i hyperbolic7 if D gt 0 ii parabolic7 if D 0 iii elliptic7 if D lt 0 Note that the type of the equation depends on the location in the plane if the coef cient a7 b7 0 depend on a and y Examples 1 For 4um3uwyum4 we get D 32 hence this equation is hyperbolic 2 For 302 um qu 0 wegetD 42y This PDE is i hyperbolic for y lt 07 and 3 7E 0 ii elliptic for y gt 07 and 3 7E 0 iii parabolic for y 07 or a 0 Finding the canonical forms Hyperbolic equations D gt 0 Choosing the transformation f yandn6y with b w b m 6 6 2c 20 we get from 21 equation U51 lower order terms 0 for UE7 o x y nvw5y In the homework you will be asked to nd a transformation such that 21 implies 055 Um lower order terms 0 Parabolic equations D 0 z The transformation 53f7 and n y gives rst aUgg lower order terms 07 1 and multiplying by 7 we get a U55 lower order terms 0 Elliptic equations D lt 0 We may assume that a gt 0 otherwise we multiply the equation by 17 this does not change D and we note that Dlt07agt0gtcgt0and2agtbE 0 Hence we can de ne the transformation 21 now implies D ZU55 Um lower order terms 07 D and multiplying by 4 7 we get 0 U55 Um lower order terms 0 Note that we can write Wk4c Wl with It should be noted that there are other transformations providing a canon ical form of the equations In particular as you are asked to show in the homework the Laplace equation is invariant under rigid transformations Examples oz um uyy 07 is called Laplace equation um uyy fac7 y 7 is called Poisson equation 5 ut k um 0 7 is called the heat equation or diffusion equation 7 utt 02 um 07 is called the wave equation where the constant c is referred to as wave speed Let us consider um c2 um 0 Introducing the transformation change of variables fct 1736 01 we get um U5 Un um U55 2U n U55 t CU U77 utt 02U 2Un Um 0 um 02 Um 402U n Hence we obtain the equivalent equation U775 0 writing 0 Un aigUn we conclude that Un is a constant function in 57 so we get U77 77 If f is a anti derivative of f we conclude that WEN 70 Q where the Q is constant with respect to 17 Hence WEN 70 g is a solution of U 717 for all possible choices of llsu iciently smooth functions f and Q and we obtain u7t 7 Ct Qct Other examples of direct integration U nU f77 Auxiliary Conditions Depending on the domain D in the plane in which the PDE is considered7 the auxiliary condition vary greatly As a rule of thumb for a well posed 5 problem we need as many auxiliary conditions as we have derivatives For functions of two variables7 that would be four in case of the hyperbolic equation and elliptic equation and three for the parabolic equation This is in particular valid for problems in bounded space domains In case of unbounded domains some times fewer conditions are necessary The most important types of conditions are the initial conditions and boundary con ditions l The initial condition usually prescribe the value of the function and its time derivative at a given time mostly for t 07 If 89 is the boundary of a domain 9 C Rquot and n is the outer normal along the boundary then the most important boundary conditions are N i u a u f Dirichlet condition 11 u a f Neumann condition 111 cm b u a f Robin condition Typical situations are the following For the hyperbolic problems I D the at plane 715 6 oo7oo gtlt oo7oo utt 02 um 0 hyp PDE IVP u70 Initial condition for U7 1143370 933 Initial condition for ut is called the initial boundary value problem for the wave equation 2 D the 7t half plane with a gt 0 33715 E 07oo gtlt 007oo um 02 um 07 hyp PDE uac7 0 Initial condition for U7 IBVP uta7 a 933 Initial condition for ut u07 t ht Boundary condition at a a and 3 D a x oo7oo 715 6 D uu 02 um 0 hyp PDE uac7 0 Initial condition for U7 1143370 933 Initial condition for ut IBVP u07 t ht Boundary condition at a 0 ub7 t gt Boundary condition at a b Both above are called initial boundary value problems IBVP Parabolic problems 1 D the 7t half plane 15gt 0 33716 E OO7OO x 07 00 ut kum 0 parab PDE IIP uac7 0 Initial condition for U7 is called the initial boundary value problem for the heat equation 2 D the rst quadrant of the 3071 plane 715 6 07 00 x 07 00 uu k um 0 parab PDE uac7 0 f Initial condition for U7 I BVP ua7 t ht Boundary condition at a a and 3 D a x 07oo 715 6 D ut k um 0 parab PDE u70 f ac Initial condition for U7 IBVP u07 t ht Boundary condition at a 0 ub7 t gt Boundary condition at a b 7 Both above are called initial boundary value problems IBVP Elliptic problems Elliptic equations with D a connected open set in the plane7 with 77smooth77 boundary 8D um uyy 0 Laplace equation D P u BD g7y Dirichlet condition is called a Dirichlet problem or boundary value problem of the rst kind um uyy f Poisson equation NP 8 a u ap gy Neumann condition n is called a Neumann problem for the Poisson equation7 or BVP of the second kind um uyy 0 Laplace equation 3 i M an b u ap i gac7 y Robin condition is called Robin problem or boundary problem of the third kind Example The initial value problem of the wave equation um 02 um 0 M 1143370 g We want to nd F and G of the general solution Mm Fa ct Go ct Hence we must have with u4m70 cFKm cwecGKmecw M G 8 g 1143370 c Fac 0 Gm CG Hence w G Fm igtdt C i gtdt 0 Fa C93 2 C33 fa gtdtC 2 Fm ms glttgtdt C and so u7t m ct fct i 7Ctgrdr 76tg7quotd7quot This we can further simplify observing that 0 g7 d7quot 7mg7quotd7quot and we get d7A1en1bert7s solution of the wave equation Z acct u7t m ct facct Jr g7 d7quot Diffusive and dispersive equations Def Solution u u7t of the form u7t Aeikx t for a linear homogeneous PDE is called a is called a plane wave i A is called the amplitude ii k is called the wave number iii w is called the temporal frequency Substituting u into the PDE provides the so called dispersion relation w wk The properties of the dispersion relation is used for some classi cation of those PDEs The equation is called dispersive7 if wk is real for real values 19 and w k 0 The equation is called diffusive7 if wk is complex for real values k Example For the PDE w mm 0 we get the dispersive equation w k3 The function wk k3 is real for real values of k and w k y 07 hence we call this equation is dispersive 10