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Register a new userAnalytical methods for solution of hypersingular and polyhypersingular integral equations
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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.
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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 optimal approximation problem, the orders of the Babenko and Kolmogorov (n-)widths of compact sets from some classes of functions have been evaluated. In conclusion we adduce some numerical illustrations for 2-D Volterra equations.
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 reformulation capable of avoiding a number of Jacobian applications is developed that, in the autonomous case, reduces the computational complexity of the algorithms. Dealing with large-scale problems, an efficient implementation based on low-rank symmetric indefinite factorizations is presented. The performance of both peer approaches up to order 4 is compared to existing implicit time integration schemes for matrix-valued differential equations.
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 for the Wiener-Hopf equation due to Henery, expressing it in terms of the Hilbert transform, and computing the latter in a more sophisticated way with sinc functions. We then generalise our method to the Fredholm equation reformulating it as two coupled Wiener-Hopf equations and solving them iteratively. We provide numerical tests and open-source code.
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 Petrov-Galerkin method with optimal test functions and prove its quasi-optimal convergence in $L^2$. Theoretical results are confirmed by numerical experiments on an open curve with uniform and adaptively refined meshes.
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 obtain asymptotic estimates of the size of small-valued sets $B_{epsilon}={xinRR : |hat{phi}(x)|leq epsilon, |x|leq R_{epsilon}}$ of Fourier transform begin{eqnarray*} hat{phi}(x)=int_{-infty}^{infty}e^{-ixt}phi (t)dt, xin RR, end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function begin{eqnarray*} p^{*}(t)=sup_{sgeq 0}[st-p(s)], tgeq 0. end{eqnarray*} Applying these results, we give an explicit estimate for the error of Tikhonovs regularization for the solution $f$ of the integral convolution equation begin{eqnarray*} int_{-infty}^{infty}f(t-s)phi (s)ds =g(t), end{eqnarray*} where $f,g in L^2(RR)$ and $phi$ is a nontrivial function of $L^1(RR)$ satisfying condition (ref{170506.1}), and $g,phi$ are known non-exactly. Also, our results extend some results of cite{tld} and cite{tqd}.
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