MATH 311 Topics in Applied Mathematics Lecture 21 Boundary value problems Separation of variables Differential equations A differential equation is an equation involving an unknown function and certain of its derivatives An ordinary differential equation ODE is an equation involving an unknown function of one variable and certain of its derivatives A partial differential equation PDE is an equation involving an unknown function of two or more variables and certain of its partial derivatives Examples x 2 2x 1 0 f 2x 2 f x 2 1 f t t 2 f t 4 u 2u u 3 u x x y y u u 5 u x y 2u 2 u u 0 0 x y algebraic equation functional equation ODE not an equation PDE functional differential equation heat equation 2u u k 2 t x wave equation 2 2u 2 u c t 2 x 2 Laplace s equation 2u 2u 0 x 2 y 2 In the first two equations u u x t In the latter one u u x y 2u 2u x 2 y 2 heat equation u k t wave equation 2u c2 2 t Laplace s equation 2u 2u 2u 0 x 2 y 2 z 2 2u 2u x 2 y 2 In the first two equations u u x y t In the latter one u u x y z Initial and boundary conditions for ODEs y t y t 0 t L General solution y t C1 e t where C1 const To determine a unique solution we need one initial condition For example y 0 1 Then y t e t is the unique solution y t y t 0 t L General solution y t C1 cos t C2 sin t where C1 C2 are constant To determine a unique solution we need two initial conditions For example y 0 1 y 0 0 Then y t cos t is the unique solution Alternatively we may impose boundary conditions For example y 0 0 y L 1 In the case L 2 y t sin t is the unique solution PDE 2u 0 w z u u w z Domain a1 w a2 b1 z b2 we allow intervals a1 a2 and b1 b2 to be infinite or semi infinite u u w z z 0 w z z Z z u w z d C w z0 u w z B z C w general solution Wave equation 2 2u 2 u c t 2 x 2 Change of independent variables w x ct z x ct How does the equation look in new coordinates w z c c t t w t z w z w z x x w x z w z 2u c2 2 t u w z 2 2 2 u u u 2 c2 w 2 w z z 2 w z 2u 2u 2u 2u 2 x 2 w 2 w z z 2 2 2 2u 2 u 2 u c 4c t 2 x 2 w z 2u Wave equation in new coordinates 0 w z General solution u x t B x ct C x ct d Alembert 1747 Boundary conditions for PDEs 2u u k 2 Heat equation t x 0 t T 0 x L Initial condition u x 0 f x where f 0 L R Boundary conditions u 0 t u1 t u L t u2 t where u1 u2 0 T R Boundary conditions of the first kind prescribed temperature u 0 t 1 t Another boundary conditions x u L t 2 t where 1 2 0 T R x Boundary conditions of the second kind prescribed heat flux u u 0 t L t 0 x x insulated boundary A particular case Robin conditions u 0 t h u 0 t u1 t x u L t h u L t u2 t x where h const 0 and u1 u2 0 T R Boundary conditions of the third kind Newton s law of cooling Also we may consider mixed boundary conditions u L t 2 t for example u 0 t u1 t x Wave equation 2 2u 2 u c t 2 x 2 0 x L 0 t T Two initial conditions u x 0 f x u x 0 g x where f g 0 L R t Some boundary conditions u 0 t u L t 0 Dirichlet conditions fixed ends Another boundary conditions u u 0 t L t 0 x x Neumann conditions free ends Linear equations An equation is called linear if it can be written in the form L u f where L V1 V2 is a linear map f V2 is given and u V1 is the unknown If f 0 then the linear equation is called homogeneous Theorem The general solution of a linear equation L u f is u u1 u0 where u1 is a particular solution and u0 is the general solution of the homogeneous equation L u 0 Linear differential operators ordinary differential operator d d2 L g0 2 g1 g2 g0 g1 g2 are functions dx dx 2 heat operator L k 2 t x 2 2 2 c t 2 x 2 a k a the d Alembertian denoted by wave operator L 2 2 Laplace s operator L 2 2 x y a k a the Laplacian denoted by or 2 How do we solve a linear homogeneous PDE Step 1 Find some solutions Step 2 Form linear combinations of solutions obtained on Step 1 Step 3 Show that every solution can be approximated by solutions obtained on Step 2 Similarly we solve a linear homogeneous PDE with linear homogeneous boundary conditions boundary problem One way to complete Step 1 the method of separation of variables Separation of variables The method applies to certain linear PDEs It is used to find some solutions Basic idea to find a solution of the PDE function of many variables as a combination of several functions each depending only on one variable For example u x t B x C t or u x t B x C t The first example works perfectly for one equation 2u t x 0 The second example proved useful for many equations Heat equation u 2u k 2 t x Suppose u x t x G t Then dG u x t dt Hence d 2 2u 2 G t x 2 dx dG d 2 x k 2 G t dt dx Divide both sides by k x G t k u x t 1 d 2 1 dG 2 kG dt dx It follows that 1 dG 1 d 2 2 const kG dt dx is called the separation constant The variables have been separated d 2 dx 2 dG dt kG Proposition Suppose and G are solutions of the above ODEs for the same value of Then u x t x G t is a solution of the heat equation Example u x t e kt sin x dG kG dt General solution G t C0 e kt C0 const d 2 dx 2 Three cases …