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Green's functions and boundary value problems Stakgold I., Holst M.
Publisher: Wiley

Ivar Stakgold's classic books "Boundary Value Problems of Mathematical Physics" or "Green's Functions and Boundary-value Problems". In this example, the a function G(x, t) which is: piecewise over a defined interval; whose pieces satisfy boundary conditions on the interval; whose pieces patch together nicely (i.e. It is easy to prove that is continuous on , here we omit it. For example, Neumann problem of Laplace equations (1),(2) is equivalent to the u to the orginal problem is to be solved and gives u through the integral formula(8). Fit together perfectly at some location t ) 's value over our defined interval. A good starting point for understanding Green's function methods is. The equation in the presence of charge is clearly more complicated and can be solved by invoking the machinery of Green's functions, which were originally directed towards electrostatic problems of this sort. Let , then the function is continuous and satisfies(1) (2). Let we call Green's function of the boundary value problem (1.1). Boundary-value problems of elliptic equations may have many different mathematical formulations, equivalent in principle but not equally efficient in practice. The kernel K has the advantage of being self-adjoint and is derived from the Green's function by double differentiation so is highly singular.