We study the existence and uniqueness of solutions for a new fractional nonlinear integro-differential equation with variable coefficients and a functional boundary condition using Krasnoselskii’s fixed point theorem and Banach’s contractive principle. The approach relies on an inverse operator in a Banach space, the Mittag-Leffler function and an implicit integral equation. The technique used has many applications to studying various nonlinear integral or differential equations, including partial differential equations, with initial or boundary conditions. Several illustrative examples are provided to show applications of our main theorems by computing approximate values of the Mittag-Leffler functions.

We carry out theoretical and numerical studies on mixed Volterra–Fredholm integral equations involving the Volterra (outer) integral of a nonlinear functional of a Fredholm (inner) integral. A contractive self-map is established and then used to prove the existence of a unique solution via the Banach contraction principle. The numerical approximation is then formulated by adopting the trapezoid rule to both the Fredholm and Volterra integrals, followed by a Newton scheme is derived to approximate the resulting nonlinear system. Next, two recently proposed Grönwall-type inequalities are applied to prove the second-order convergence of our proposed algorithm. Examples are provided to assess the performance of the method, and the results verify the second order convergence and accuracy of our proposed method.

We propose the class of Fredholm–Hammerstein type integral equations by considering three different types of weakly singular kernels, such as the Hadamard type kernel, the mixed type kernel, and the algebraic kernel. As the Hadamard type kernel and mixed type kernel are the combination of |x−s|−α, 0<α<12 and (log|x−s|)n, n∈ℕ, they are therefore difficult to handle. Moreover, the algebraic kernel contains singularities at two points. Furthermore, due to the presence of singularities inside the domain, it isn’t easy to analyze the solution. We have used fixed-point iteration to study the existence of solutions. Additionally, we have demonstrated that the solution is unique to the particular range of the parameters. The theoretical results are validated using numerical examples.

We investigate time-optimal controls invoked by Sobolev-type multiterm time fractional differential equations in new forms. We prove the existence of mild solutions of Sobolev type multiterm time-fractional evolution equations in the Caputo and Riemann–Liouville fractional derivatives; then we investigate time-optimal control of such systems. Our results fall under the framework of the resolvent operator family related to the Sobolev-type fractional evolution equations. The uniqueness of solutions and the reflexivity of the state-valued space are not required for time-optimal control problems.

A new generalized American option under ψ-Caputo fractional-order derivative Heston (AOψCFDH) model is investigated. A new numerical implicit scheme method (NISM) has been developed for solving the AOψCFDH model. The stability and convergence of the NISM are analyzed. Two numerical examples are proposed in order to show the robustness and the efficiency of both the model AOψCFDH and the NISM.

We study the existence of a unique bounded continuous solution of a Caputo fractional differential equation. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular, and hence the standard techniques that are normally used for Volterra integral equations do not apply here. This hurdle is overcome by using a resolvent equation, applying some known properties of the resolvent, and then employing the contraction mapping principle.

We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory behaviors due to the oscillation of the kernel. It was pointed out recently that standard DNNs favour low frequency functions, and as a result, they often produce poor approximation for functions containing high frequency components. We addressed this issue in this study. We first developed a numerical method for solving the equation with DNNs as an approximate solution by designing a numerical quadrature that tailors to computing oscillatory integrals involving DNNs. We proved that the error of the DNN approximate solution of the equation is bounded by the training loss and the quadrature error. We then proposed a multi-grade deep learning (MGDL) model to overcome the spectral bias issue of neural networks. Numerical experiments demonstrate that the MGDL model is effective in extracting multiscale information of the oscillatory solution and overcoming the spectral bias issue from which a standard DNN model suffers.

We study compact intertwining relations for composition operators whose intertwining operators are Volterra-type operators between Morrey spaces and Bloch-type spaces in the unit disk. As a consequence, we find a new connection between Morrey spaces and Bloch-type spaces through these relations.

We study the existence of a strong solution for a class of second-order nonlinear retarded differential equations involving a lower-order fractional term. We prove the existence result by using Rothe's time-discretization method. We also provide some examples to examine the existence of solutions for a class of second-order fractional differential equations.

While an integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions is available in the literature, a proof of this formula seems to be missing. Moreover, the available formula contains an integral term including the time derivative of the fundamental solution of the heat equation, whose interpretation is difficult at second glance. To fill these gaps, we provide a rigorous proof of a general version of the integration by parts formula and an alternative representation of the mentioned integral term, which is valid for a certain class of functions including the typical tensor-product discretization spaces.