We study the spatially inhomogeneous coagulation condensation process with multiple fragmentations, proving the existence of a continuous solution of the model equation corresponding to the coagulation kernel

$$K(x,y)\le k_0+k_1(x^{\alpha }+y^{\alpha }) \text{for} x,y\in [0,\infty ), \text{where} \alpha \in \left[0,1\right],k_0,k_1\ge 0,$$

with at least one of $k_0$ and $k_1$ is nonzero. This form of the coagulation kernel includes several practical kernels. The study includes the fragmentation part in the model equation and considers multiple fragmentation kernels. Finally, the uniqueness of the solution is also investigated.

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 favor 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 multigrade 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.

This paper discusses a multiterm time fractional delay differential equation in a real Hilbert space. An iterative scheme for a multiterm time fractional differential equation is established using Rothe’s method. The method of semidiscretization is extended to this kind of time fractional problem with delay in the case that the time delay parameter $\nu >0$ satisfies $\nu \le T$, where $T$ denotes the final time. We apply the accretivity of the operator $A$ in an iterative scheme to establish the existence and regularity of strong solutions to the considered problem. Finally, an example is provided to demonstrate the abstract result.

We establish new inequalities of generalized Ostrowski- and Ostrowski–Grüss-type on fractal sets. They depend on a new generalized Montgomery identity on fractal sets and a parameter function $h:[0,1]\to [0,1]$. Our results not only give an error estimate for $\frac{\mathrm{\Gamma }(1+\alpha )}{{(b-a)}^{\alpha }}{}_a{I}_b^{\alpha }f(t)$ with $f(x)$, $f(a)$ and $f(b)$, but also generalize and extend previous results of Sarikaya, Budak and coauthors. We give applications to two-point and three-point quadrature formulas.

We consider the scattering of time-harmonic electromagnetic waves by a bounded, penetrable, homogeneous obstacle. This problem admits an equivalent formulation in terms of a strongly singular volume integral equation (VIE). In this paper, and for smooth interfaces, we construct a regularizer for the operator that describes the VIE, i.e., we give an explicit representation of an integral operator, which, applied to the VIE, transforms it into the form “identity plus a compact operator”. The employed strategy is inspired by the previous work (Costabel et al. 2012).

In the literature, asymptotic expansions for certain approximate solutions of operator equations and eigenvalues associated with Green’s kernel are not justified in the case of iterated collocation method for piecewise polynomial space of degree $\ge 2$. In this paper, we prove the existence of asymptotic expansion of operator equations at partition points and for a simple eigenvalue associated with Green’s type kernel. A numerical example is considered to illustrate theoretical results.