"Boundary value problem, complex-variable methods", Encyclopedia of Mathematics, EMS Press, 2001."Boundary value problems in potential theory", Encyclopedia of Mathematics, EMS Press, 2001.Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ^ ISO/IEC/IEEE International Standard - Systems and software engineering.These categories are further subdivided into linear and various nonlinear types.Īpplications Electromagnetic potential For a hyperbolic operator, one discusses hyperbolic boundary value problems. For an elliptic operator, one discusses elliptic boundary value problems. įor example, if the independent variable is time over the domain, a boundary value problem would specify values for y ( t ) Īside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. A boundary value is a data value that corresponds to a minimum or maximum input, internal, or output value specified for a system or component. A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value). Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.Īmong the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation) the solution was given by the Dirichlet's principle.īoundary value problems are similar to initial value problems. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. To be useful in applications, a boundary value problem should be well posed. The analysis of these problems involves the eigenfunctions of a differential operator. A large class of important boundary value problems are the Sturm–Liouville problems. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.īoundary value problems arise in several branches of physics as any physical differential equation will have them. In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. Where Q is the flow domain and f3 is a known function.Shows a region where a differential equation is valid and the associated boundary values Typically, the initial condition can be expressed by h(x,y,Q)=Mx,y) (x,y)eQ (21) Prior to solving the groundwater flow equation, the initial condition must be specified in the flow domain. The Cauchy, or mixed boundary, condition occurs when both the head and its normal derivative are specified on the boundary 9fi3, for example, the induced infiltration in a coupled aquifer and stream system (Prickett and Lonnquist, 1971). Where f2 is a known function and dh/dn is the normal derivative to the boundary 9il2-ģ. For example, if a portion of the aquifer boundary is subject to recharge, then Neumann conditions occur when a portion of the boundary has a specified flow transversing it normal to the boundary. Where/j is a known function and 9ilj is the boundary.Ģ. For example, if an aquifer is adjacent to a stream or lake, then h(x,y, t) = /j(x, y, t) (x, e 9Qt Dirichlet conditions occur when a portion of the boundary is at a prescribed head level. The most common types of boundary conditions are Dirichlet, Neumann, and Cauchy.ġ.
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