Analysis of extended boundary-domain integral and integro-differential equations of some variable-coefficient BVP

For a function from the Sobolev space H1(Ω) definitions of non-unique external and unique internal co-normal derivatives are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain Ω, where they are prescribed, to the domain boundary, where they are not. The notions are then applied to formulation and analysis of direct boundary-domain integral and integro-differential equations (BDIEs and BDIDEs) based on a specially constructed parametrix and associated with the Dirichlet boundary value problems for the "Laplace" linear differential equation with a variable coefficient and a rather general right hand side. The BDI(D)Es contain potential-type integral operators defined on the domain under consideration and acting on the unknown solution, as well as integral operators defined on the boundary and acting on the trace and/or co-normal derivative of the unknown solution or on an auxiliary function. Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP are investigated in appropriate Sobolev spaces

Unique solvability of boundary value problem for functional differential equations with involution

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established

Unique solvability of boundary value problem for functional differential equations with involution

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K ˜2( t,s ) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established

Regularized BIE formulations for first- and second-order shape sensitivity of elastic fields.

The subject of this paper is the formulation of boundary integral equations for first- and second-order shape sensitivities of boundary elastic fields in three-dimensional bodies. Here the direct differentiation approach is considered. It relies on the repeated application of the material derivative concept to the governing regularized (i.e. weakly singular) displacement boundary integral equation (RDBIE) for an elastostatic state on a given domain. As a result, governing BIEs, which are also weakly singular, are obtained for the elastic sensitivities up to the second order. They are formulated so as to allow a straightforward implementation; in particular no strongly singular integral is involved. It is shown that the actual computation of shape sensitivities using usual BEM discretization uses the already built and factored discrete integral operators and needs only to set up additional right-hand sides and additional backsubstitutions. Some relevant discretization aspects are discussed

Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates

We obtain a priori local pointwise second derivative estimates for solutions $u$ to a class of augmented Hessian equations on Riemannian manifolds, in terms of the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any $W^{2,p}$ norm can be dropped. Our results are derived using integral estimates

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient $A$ and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of $A$, fractional order $\alpha$ and the smoothness of the first initial condition, as well as to the properties of the equation's right-hand side $f(t)$. The resulting method possesses exponential convergence for positive sectorial $A$, any finite $t$, including $t = 0$, and the whole range $\alpha \in (0,2)$. It is suitable for a practically important case, when no knowledge of $f(t)$ is available outside the considered interval $t \in [0, T]$. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates

Finding the zeros of ahlfors map using integral equation method on bounded multiply connected regions

The Ahlfors map of an n-connected region is a n-to-one map from the region onto the unit disk. The Ahlfors map being n-to-one map has n zeros. Previously, the exact zeros of the Ahlfors map are known only for the annulus region and a particular triply connected region. The zeros of the Ahlfors map for general bounded multiply connected regions has been unknown for many years. The purpose of this research is to find the zeros of the Ahlfors map for general bounded multiply connected regions using integral equation method. This work develops six new boundary integral equations for Ahlfors map of bounded multiply connected regions. The kernels of these integral equations are the generalized Neumann kernel, adjoint Neumann kernel, Neumann-type kernel and Kerzman-Stein type kernel. These integral equations are constructed from a non-homogeneous boundary relationship satisfied by an analytic function on a multiply connected region. The first four integral equations have kernels containing the zeros of the Ahlfors map which are unknown. The fifth integral equation has no zeros of the Ahlfors map in the kernel but involves derivative of the Ahlfors map at the unknown zeros. The sixth integral equation has unknown zeros appearing only at the right-hand side. The sixth integral equation proves to be useful for computing the zeros of the Ahlfors map. This work presents a numerical method for computing the zeros of Ahlfors map of any bounded multiply connected region with smooth boundaries. This work derives two formulas for the derivative of the boundary correspondence function of the Ahlfors map and the derivative of the Szeg¨o kernel. The relation between the Ahlfors map and the Szeg¨o kernel is classical. The Szeg¨o kernel is a solution of a Fredholm integral equation of the second kind with the Kerzman-Stein kernel. These formulas are then used along with the sixth integral equation to compute all the zeros of the Ahlfors map for any bounded smooth multiply connected regions. Some examples are presented to demonstrate the efficiency of the presented method