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lecture23 - Dierential Equations Linear Algebra Lecture 23...

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Differential Equations & Linear Algebra Lecture 23 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010

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Contents Two-point boundary value problems
Contents Two-point boundary value problems Eigenvalue problems

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Contents Two-point boundary value problems Eigenvalue problems Fourier series
Partial differential equations In many important physical problems there are two or more independent variables, so the corresponding mathematical models involve partial, rather than ordinary, differential equations. One important method for solving PDEs is separation of variables. This method essentially transforms the PDE into a set of ordinary differential equations. Usually the solution will be then expressed as an infinite series. In many cases, we ultimately need to deal with a series of sines and/or cosines, or Fourier series. We will mainly study two simple equations arising from heat conduction and vibration of an elastic string.

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Initial conditions and boundary conditions Up to this point we have dealt with initial value problems, consisting of a differential equation together with suitable initial conditions at a given point. A typical example is y + p ( t ) y + q ( t ) y = g ( t ) , (1) with initial conditions y ( t 0 ) = y 0 , y ( t 0 ) = y 0 . (2) Physical applications often lead to another type of problem, in which the value of the dependent variable or its derivative is specified at two different points. Such conditions are called boundary conditions .
Two-point boundary value problems A differential equation and suitable boundary conditions form a two-point boundary value problem . A typical example is y + p ( x ) y + q ( x ) y = g ( x ) , (3) with boundary conditions y ( α ) = y 0 , y ( β ) = y 1 . (4) The solution of the boundary value problem (3), (4) is a function y = φ ( x ) that satisfies both (3) in the interval α < x < β and (4). Usually, we first seek the general solution of the differential equation and then use the boundary conditions to determine the values of the arbitrary constants.

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No existence and uniqueness for solutions If the function g ( x ) = 0 for all x and if the boundary values y 0 = y 1 = 0 , then the boundary value problem (3), (4) is called homogeneous . Otherwise, the problem is nonhomogeneous . Although the initial value problem and the boundary value problem may appear to be quite similar, their solutions differ in some very important ways. Under mild conditions, initial value problems are certain to have a unique solution. On the other hand, boundary value problems under similar conditions may have a unique solution, but they may also have no solution, or in some cases, have infinitely many solutions. A homogeneous problem always has at least one solution y = 0 .
Example Example Solve the boundary value problem y + 2 y = 0 , y (0) = 1 , y ( π ) = 0 . (5) Solution The general solution of the differential equation (5) is y = c 1 cos 2 x + c 2 sin 2 x. (6) Plugging in boundary conditions for (6), we have c 1 = 1 , c 2 = - cot 2 π . Thus the solution of the boundary value problem (5) is y = cos 2 x - cot 2 π sin 2 x.

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lecture23 - Dierential Equations Linear Algebra Lecture 23...

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