We consider the maximal regularity problem for parabolic and hyperbolic nonautonomous evolution equations in Hilbert space damped by some nonlocal time-fractional derivative with time-dependent variable-order. We prove the existence, uniqueness and other regularity properties of the solutions of these problems under minimal regularity assumptions on the operators, initial data and inhomogeneous terms.

We analyze a nonlinear Klein–Gordon–Schrödinger system in ${ℝ}^n\times {ℝ}^{+}$, $n\ge 1$, in a space-time fractional setting, considering the time fractional variation in the Caputo sense, and a fractional spatial dispersion. Assuming general polynomial nonlinearities, we prove the existence of local and global mild solutions, as well as the asymptotic stability of global mild solutions, with initial data in a large class of singular spaces, namely, the weak $L^p$ spaces. We also derive the existence of local and global solutions in the same framework for the nonlinear time-space fractional Klein–Gordon equation.

Often in scattering theory, two-dimensional singular integrals arise over a circular or elliptic region. So far, formulas either for evaluating Cauchy principal value integrals over circular and elliptic regions or for computing hypersingular integrals over circular region are available. Thus, in the present paper, we derive a formula that evaluates the dominant part of a hypersingular integral equation that appears in a variety of physical problems involving an elliptic disc in terms of trigonometric and associated Legendre functions $P_{\nu }^{\mu }$. Expanding the integrand function in terms of Fourier series in the azimuthal direction, orthogonal polynomials in the radial variable, and the Bessel function $J_{\nu }$ in terms of $P_{\nu }^{\mu }$, utilizing recurrence relations for $P_{\nu }^{\mu }$ and $J_{\nu }$, and incorporating the value of the Weber–Schafheitlin integral, we derive an expression for the hypersingular integral in terms of hypergeometric functions. Using relationships of hypergeometric functions and their connections with $P_{\nu }^{\mu }$, finally the dominant integral is expressed in terms of $P_{\nu }^{\mu }$. This result generalizes previously known integral identities which are applicable to weakly singular kernels on elliptic disc or to hypersingular integrals on circular discs. The formula obtained is verified by recovering known results for the integral over a circular disc. In particular, the added mass due to heaving circular and elliptic discs in an unbounded fluid are recovered. A spectral method for the solution of a class of hypersingular integral equations over an elliptic disc is proposed and numerical results for a simple case are presented, showing the numerical convergence of the method.

We studied a new class of nonlinear infinite system of functional integral equations with Riemann–Liouville fractional operator and their existing solution in the sequence space ${\ell }_p$, $p>1$. We first prove the existence of a solution for the infinite system of functional integral equation by using Hausdorff measure of noncompactness and the generalized Meir–Keeler fixed-point theorem. Also, we have presented an example to illustrate the effectiveness of our main result. Further, we propose an iterative algorithm formed by homotopy perturbation along with the Adomian decomposition method to solve the considered problem with acceptable accuracy, which converges strongly to the approximate solution.

We proposed a result which generalizes the Darbo’s fixed-point theorem (DFPT) by using measure of noncompactness (MNC). Then, we use DFPT to investigate the existence of the solution of generalized proportional fractional integral equation (FIE) of two variables. Then we present a suitable example to prove the reliability of the method. Also, we use an iterative algorithm based on the modified homotopy perturbation (MHP) method and the sinc interpolation to estimate the solution of the presented example with sufficient precision, and introduce the application of the obtained solution in industry.

Stochastic integral equations are used to model a variety of phenomena in science and engineering, including finance, physics and biology. They are often used to model systems that involve randomness or uncertainty, such as stock prices, weather patterns, or biological populations. Solving stochastic integral equations can be challenging, as the integral involves a random process. There are various techniques that can be used to solve such equations, such as the Wiener–Itô integral, the Stratonovich integral and the Malliavin calculus. These techniques involve manipulating the integral in various ways, such as approximating it with a Riemann sum, using Taylor expansions, or applying the Itô or Stratonovich rules. In this work, the stochastic integral equations has been solved using exponential linear unit-based neural networks. Monte Carlo based approach has been implemented to discretize the integral equation and stochastic gradient descent techniques have also been implemented to optimize the loss function discussed. The numerical results and convergence analysis have been discussed for various cases to justify the versatility of our proposed schemes. Loss function graphs have also been discussed for more clarity on the behavior of the solutions based on our proposed technique. With the use of exponential linear unit as the activation function, the convergence was smooth and stable. It has been shown in the results that the proposed methodology is able to predict the exact solution for a stochastic integral equation at an average loss of 0.01.

We solve the longstanding positivity problem of solutions to Caputo fractional-order SEIR models. The issues of existing results are introduced first. Then the rationale for the positivity problem is explored and found to be the nonlocality of Caputo fractional differential operators. Benefiting from Vainikko’s definition and our continuation theorem, the positivity problem is exactly solved. Finally, numerical simulations are provided to illustrate the results.