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We propose a method for transformating linear and nonlinear hypersingular integral equations into ordinary differential equations. Linear and nonlinear polyhypersingular integral equations are transformed into partial differential equations. Well known that many types of differential equations can be solved in quadratures. So, we can receive analytical solutions for many types of linear and nonlinear hypersingular and polyhypersingular integral equations.
Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely related with the
We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type schemes, a r
We present numerical methods based on the fast Fourier transform (FFT) to solve convolution integral equations on a semi-infinite interval (Wiener-Hopf equation) or on a finite interval (Fredholm equation). We extend and improve a FFT-based method fo
We present an ultra-weak formulation of a hypersingular integral equation on closed polygons and prove its well-posedness and equivalence with the standard variational formulation. Based on this ultra-weak formulation we present a discontinuous Petro
Let $phi$ be a nontrivial function of $L^1(RR)$. For each $sgeq 0$ we put begin{eqnarray*} p(s)=-log int_{|t|geq s}|phi (t)|dt. end{eqnarray*} If $phi$ satisfies begin{equation} lim_{sto infty}frac{p(s)}{s}=infty ,label{170506.1} end{equation} we obt