Transcription of Chapter 5 Boundary Value Problems
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Chapter 5 Boundary Value Problems
Chapter 5 Boundary Value ProblemsA Boundary Value problem for a given differential equation consists of finding a solution of thegiven differential equation subject to a given set of Boundary conditions . A Boundary conditionis a prescription some combinations of values of the unknownsolution and its derivatives at morethan one (a, b) Rbe an interval. Letp,q,r: (a, b) Rbe continuous this Chapter we consider the linear second orderequation given byy +p(x)y +q(x)y=r(x), a < x < b.( )Corresponding to ODE ( ), there are four important kinds of (linear) Boundary conditions . Theyare given byDirichlet or First kind :y(a) = 1, y(b) = 2,Neumann or Second kind :y (a) = 1, y (b) = 2,Robin or Third or Mixed kind : 1y(a) + 2y (a) = 1, 1y(b) + 2y (b) = 2,Periodic :y(a) =y(b), y (a) =y (b).Remark (On periodic Boundary condition)If the coefficients of ODE( )are periodicfunctions with periodl=b aand if is a solution of ODE( )(note that this solution existsonR), then defined by (x) = (x+l)is also a solution.
Boundary Value Problems A boundary value problem for a given differential equation consists of finding a solution of the given differential equation subject to a given set of boundary conditions. A boundary condition is a prescription some combinations of values of the unknown solution and its derivatives at more than one point.
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