Abstract
On the basis of the exact solution of biharmonic problems of elasticity theory in a half-strip one possible reason is shown of those problems that arise when an approximate or numerical approaches leading the solution of boundary value problems to infinite systems of linear algebraic equations. Construction of exact solutions of some boundary value problems for differential equations in partial derivatives is not possible without their extensions to Riemann surfaces. Moreover, each of the boundary value problem corresponds to its Riemann surface. This fact is important to consider when developing an effective approximate and numerical methods of solving boundary value problems.
Highlights
In the articles [1] [2] the theory was developed and the examples of exact solutions of biharmonic problem of elasticity theory in a half-strip and in a rectangle was first obtained
On the basis of the exact solution of biharmonic problems of elasticity theory in a half-strip one possible reason is shown of those problems that arise when an approximate or numerical approaches leading the solution of boundary value problems to infinite systems of linear algebraic equations
Nonuniqueness is associated with non-finite of biorthogonal functions and, as a consequence, the necessity of continuation the given at the end face of the halfstrip boundary functions from the segment to the whole real axis
Summary
In the articles [1] [2] the theory was developed and the examples of exact solutions of biharmonic problem of elasticity theory in a half-strip and in a rectangle was first obtained (an overview of biharmonic problem for almost 200 years is given in the article [3]). There is the solution of the principal boundary value problem of elasticity theory in a half-strip. 2. The Solution of the Boundary Value Problem of Elasticity Theory in a Half-Strip. Consider the solution of boundary value problem of elasticity theory for the biharmonic equation in the half-strip {Π : x ≥ 0,| y |≤ 1} , the long sides of which are free, i.e. stresses are:. Satisfying using expressions (2.3) that given at the end of the half-strip the normal σ ( y) and tangential τ ( y) stresses, we come to the problem of determining the coefficients ak , ak from the boundary conditions ( ) ( ) ∞. Desired expansion coefficients are determined from the expansions (2.6)
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