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Contribution to the Theory of Upper and Lower Bounds in Boundary Value Problems

1955; Physical Society of Japan; Volume: 10; Issue: 1 Linguagem: Inglês

10.1143/jpsj.10.1

1347-4073

Hiroshi Fujita,

Numerical methods for differential equations

In the present paper certain approximate methods concerning self-adjoint operators H of Hilbert spaces which can be expressed in the form H = T * T are developed. Some theorems are formulated with remarks and examples. They give a method of obtaining upper and lower bounds of the value of the solution, at each preassigned point, of a boundary value problem. As a numerical example we consider the following Poisson's equation, -Δ u =1 in D, u =0 on C, where D denotes the square domain in the x y -plane with vertices at (1, -1), (1, 1), (-1, 1), (-1, -1) and C is the boundary of this square. Upper and lower bounds of the value of the solution u at the origin are calculated. The result of the second approximation is 0.2939≦ u (0, 0)≦0.2953. In order to show that the results are applicable to many problems, several differential operators familiar in mathematical physics, for instance, Laplace's differential operators with different boundary conditions are reduced to the form T * T .