For an inverse nonlinear diffusion equation with conformable time derivative, we study the ill-posed property in the sense of Hadamard. To obtain a stable numerical solution, we propose two regularization methods. The results of existence and uniqueness, regularity and stability of the regularized problem are obtained. We also show that the corresponding regularized solutions converge to the sought solution strongly in $L^2\left({ℝ}^d\right)$ uniformly under some a priori assumptions on the solution.

We analyze the existence and uniqueness of solutions in $L^p$ spaces to a class of integral equations of the third kind with weakly singular kernels.

We consider a linear Volterra integral equation of the second kind with a sum kernel $K\left(t^{\prime },t\right)={\sum }_i{K}_i\left(t^{\prime },t\right)$ and give the solution of the equation in terms of solutions of the separate equations with kernels $K_i$, provided these exist. As a corollary, we obtain a novel series representation for the solution with improved convergence properties. We illustrate our results with examples, including the first known Volterra equation solved by Heun’s confluent functions. This solves a long-standing problem pertaining to the representation of such functions. The approach presented here has widespread applicability in physics via Volterra equations with degenerate kernels.

We derive a fractional multiterm Gronwall lemma and use it to study the stability of the Cauchy problem for a general fractional integrodifferential system of Volterra type.

We study the time-fractional parabolic integrodifferential equations with the Caputo fractional time derivative of order $\alpha \in \left(0,1\right)$ in a bounded convex polygonal domain in ${ℝ}^d$. We prove the existence and uniqueness of the solution using the eigenfunction expansion and establish a priori bounds for the solution under various regularity assumptions on the initial data and the source function. Our study includes the initial data in the spaces ${Ḣ}^2\left(\mathrm{\Omega }\right)$, $H_0^1\left(\mathrm{\Omega }\right)$ and $L^2\left(\mathrm{\Omega }\right)$ while the source function belongs to the class of Hölder continuous and bounded functions. It is shown that the solution of the corresponding homogeneous problem is infinitely differentiable with respect to time $t$ when the initial function is an element of $L^2\left(\mathrm{\Omega }\right)$. Finally, we derive a general stability results for the solution of the homogeneous problem.

We study the existence of mild solutions to subdiffusion equations with memory

$$\left(\ast \right){\partial }_t^{\alpha }u\left(t\right)=Au\left(t\right)+{\int }_0^t\kappa \left(t-s\right)Au\left(s\right)ds,t\ge 0,$$

with the initial condition $u\left(0\right)=x$, where $0<\alpha <1$, $A$ is a closed linear operator defined on a Banach space $X$, the initial value $x$ belongs to $X$ and $\kappa$ is a suitable kernel in $L_{loc}^1\left({ℝ}_{+}\right)$. First, we find a subordination formula for the solution operator of $\left(\ast \right)$ and then we study its connection with the existence of mild solution to the first order diffusion equation with memory.

We consider a Fredholm integral equation of the second kind with kernel of the type of Green’s function. Iterated Galerkin method is applied to such an integral equation. For $r\ge 1$, a space of piecewise polynomials of degree $\le r-1$ with respect to a uniform partition is chosen to be the approximating space. We obtain an asymptotic expansion for the iterated Galerkin solution at the partition points. Richardson extrapolation is used to increase the order of convergence. A numerical example is considered to illustrate our theoretical results.