The mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The PDE on the right-hand side belongs to $H^{-1}\left(\mathrm{\Omega }\right)$ or ${\tilde{H}}^{-1}(\text{Ω})$, when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials, and hence the unique solvability of BDIEs. The equivalence of the BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the corresponding Sobolev spaces.

Using the theory of Young measures, we prove the existence of weak solutions to the following quasilinear elliptic system:

$$A\left(u\right)=f\left(x\right)+\text{div }{\sigma }_0\left(x,u\right),$$

where $A\left(u\right)=-\text{div }\sigma \left(x,u,Du\right)$ and $f\in W^{-1}{L}_{\overline{M}}\left(\mathrm{\Omega };{ℝ}^m\right)$. This problem corresponds to a diffusion phenomenon with a source $f$ in a moving and dissolving substance, where the motion is described by ${\sigma }_0$.

We study the unique continuation property for the $N$-dimensional BBM equations using Carleman estimates. We prove that if the solution of this equation vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.

We present the approximate analytical method of construction for the complex potential of the locally sourceless, locally irrotational plane flow in a multiply connected infinite domain. The velocity at infinity, the circulations around the impermeable boundary components and the fluxes across the permeable boundary components are given. The method is based on reduction of Fredholm integral equation to the linear system. This method is easily computable.

We study the uniqueness of solutions for certain partial integro-differential equations with the initial conditions in a Banach space. The results derived are new and based on Babenko’s approach, convolution and Banach’s contraction principle. We also include several examples for the illustration of main theorems.

We establish some new weakly singular nonlinear integral inequalities, which generalize some known integral inequalities. The inequalities given here can be used in the analysis of the qualitative properties of fractional differential equations and integral equations. Applications are also provided to illustrate the usefulness of our results.

This work is devoted to the solvability of the weighted Cauchy problem for fractional differential equations of arbitrary order, considering the Riemann–Liouville derivative. We show the equivalence between the weighted Cauchy problem and the Volterra integral equation in the space of Lebesgue integrable functions. Finally, we point out some discrepancies between the solutions for fractional and integer order case.

In this paper, a fast Fourier–Galerkin method is presented for solving a system of integral equations, which is a reformulation of the Dirichlet problem of the biharmonic equation. This method is based on operator splitting and truncation strategy designing. The truncated matrix has only $\mathcal{𝒪}\left(n\text{ log }n\right)$ nonzero entries, but the approximate solutions preserve the stability and optimal convergence order. Numerical examples indicate the theoretical estimate.