The weak-form of Helmholtz differential equation, in conjunction with vector test-functions (which are gradients of the fundamental solutions to the Helmholtz differential equation in free space) is utilized as the basis in order to directly derive non-hyper-singular boundary integral equations for the velocity potential, as well as its gradients. Thereby, the presently proposed boundary integr...

Applications of a new and efficient composite BIE formulation for 3-D acoustic problems are studied in this paper, based on an improved weakly-singular form of the hypersingular boundary integral equation (HBIE). This new form of the HBIE involves only tangential derivatives of the density function and thus its discretization using the boundary element method (BEM) is easier to perform. C conti...

We discuss a possibility to construct high order methods on uniform or mildly graded grids for the numerical solution of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. Using an integral equation reformulation of the initial value problem, we apply to it a smoothing transformation so that the exact solution of the resulting equation does not conta...

Fractional directional integrals are the extensions of the Riemann-Liouville fractional integrals from oneto multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as Newton-Cotes and Gauss-Legendre rules. It ...

Abstract In this paper we first establish the existence, uniqueness and Hölder continuity of solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, singularities $\alpha \in (0, 1)$ for drift term $\beta 1/2)$ term. Subsequently, propose a $\theta $-Euler–Maruyama scheme Milstein solve numerically obtain strong rates convergence both schemes in $L^{p}$ norm any...