Abstract
We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today, hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approximate equations system in terms of the operator logarithmic norm. This makes it possible to estimate the norm of the inverse matrix for an approximating system. The advantage of the method for nonlinear equations is that neither the existence or reversibility of the nonlinear operator derivative is required. Examples are given illustrating the effectiveness of the proposed method.
Highlights
The importance of developing analytical and numerical methods for solving hypersingular integral equations is determined by a variety of fields of mathematics and by applications that use hypersingular integral equations
The boundary integral equations method reduces the dimensions of partial differential equations; that leads to hypersingular integral equations
Hypersingular integral equations, singular integral equations and Riemann boundary problem are widely used in aerodynamics, electrodynamics, quantum physics, antennae theory and many other fields of physics and engineering [1,2,3,4,5]
Summary
The importance of developing analytical and numerical methods for solving hypersingular integral equations is determined by a variety of fields of mathematics and by applications that use hypersingular integral equations. In [21,22] spline-collocation methods for solving hypersingular and polyhypersingular integral equations of the second kind with odd and even singularities have been developed and justified. For hypersingular integral equations with bounded right-hand sides the convergence of an approximate solution sequence to the exact solution was proven under certain additional conditions. The problem arises of constructing and justifying approximate methods for solutions for hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We propose an approach to solving linear and nonlinear hypersingular integral equations, the right parts of which contain functions with power features.
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