Integration Of Differential Equations Research Articles

I/O-efficient iterative matrix inversion with photonic integrated circuits

Photonic integrated circuits have been extensively explored for optical processing with the aim of breaking the speed and energy efficiency bottlenecks of digital electronics. However, the input/output (IO) bottleneck remains one of the key barriers. Here we report a photonic iterative processor (PIP) for matrix-inversion-intensive applications. The direct reuse of inputted data in the optical domain unlocks the potential to break the IO bottleneck. We demonstrate notable IO advantages with a lossless PIP for real-valued matrix inversion and integral-differential equation solving, as well as a coherent PIP with optical loops integrated on-chip, enabling complex-valued computation and a net inversion time of 1.2 ns. Furthermore, we estimate at least an order of magnitude enhancement in IO efficiency of a PIP over photonic single-pass processors and the state-of-the-art electronic processors for reservoir training tasks and multiple-input and multiple-output (MIMO) precoding tasks, indicating the huge potential of PIP technology in practical applications.

Multivalued nonlinear dominated mappings on a closed ball and associated numerical illustrations with applications to nonlinear integral and fractional operators

In the context of complete strong b-metric-like spaces, we prove new multi-fixed point solutions for the pair of multivalued, dominated operators that fulfill the generalized nonlinear contractions on a closed ball. We employ a mix of two different types of mappings in our approach: one is a class of multi-dominated mappings, while the other is a weaker class of strictly increasing mappings. Additionally, some new fixed-point results concerning the multi-graph-dominated structure in graph contraction are presented. To validate the hypothesis of acquired results, a few sample cases are presented. A numerical experiment has been carried out to approximate the fixed point. In addition, to demonstrate the originality of our findings, we proposed simple and efficient solutions to the system of fractional differential equations and nonlinear Volterra-type integral equations.

Computational and analytical analysis of integral-differential equations for modeling avoidance learning behavior

AbstractThis work emphasizes the computational and analytical analysis of integral-differential equations, with a particular application in modeling avoidance learning processes. Firstly, we suggest an approach to determine a unique solution to the given model by employing methods from functional analysis and fixed-point theory. We obtain numerical solutions using the approach of Picard iteration and evaluate their stability in the context of minor perturbations. In addition, we explore the practical application of these techniques by providing two examples that highlight the thorough analysis of behavioral responses using numerical approximations. In the end, we examine the efficacy of our suggested ordinary differential equations (ODEs) for studying the avoidance learning behavior of animals. Furthermore, we investigate the convergence and error analysis of the proposed ODEs using multiple numerical techniques. This integration of theoretical and practical analysis enhances the domain of applied mathematics by providing important insights for behavioral science research.

Valuations of generalized variance swaps under the jump–diffusion model with stochastic liquidity risk

This paper focuses on the pricing problem of generalized variance swaps with jump risk in the underlying asset price under a stochastic liquidity model. We obtain a pricing formula of generalized variance swap in a jump–diffusion model with stochastic liquidity risk by the joint moment generating function (MGF) generated by solving the partial integral differential equation (PIDE). Using asymptotic analysis, we also demonstrate that as the sampling interval approaches zero, the pricing formula of discretely sampled generalized variance swap tends to be that of continuously sampled generalized variance swap. Finally, to verify the feasibility of the pricing formula of the generalized variance swap presented in this paper, we conduct some numerical experiments, including a comparison with the results of Monte Carlo (MC) simulation, the impact of various model parameters on the delivery prices of generalized variance swaps, and empirical research using actual market data.

The Solvability and Explicit Solutions of Singular Integral–Differential Equations with Reflection

This article deals with a classes of singular integral–differential equations with convolution kernel and reflection. By means of the theory of boundary value problems of analytic functions and the theory of Fourier analysis, such equations can be transformed into Riemann boundary value problems (i.e., Riemann–Hilbert problems) with nodes and reflection. For such problems, we propose a novel method different from classical one, by which the explicit solutions and the conditions of solvability are obtained.

Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment

In order to deal with complex risk scenarios involving claims, uncertainty, and investments, we consider the ruin problems in a compound Poisson risk model with liquid reserves and proportional investments and study the expected discounted penalty function under threshold dividend strategies. Firstly, the integral differential equation of the expected discounted penalty function is derived. Secondly, since the closed-form solution of the equation cannot be obtained, a sinc method is used to obtain the numerical approximation solution of the equation. Finally, the feasibility and superiority of the sinc method are illustrated by error analysis. In addition, based on a symmetric jump risk market, we discuss the influence of some parameters on the ruin probability with some examples. This study can help actuaries develop more robust risk management strategies and ensure the long-term stability and profitability of insurance companies. It provides a theoretical basis for actuaries to carry out risk management.

Generalizations of Riemann–Liouville fractional integrals and applications

Generalizations of Riemann–Liouville fractional integrals and applications

On the interfacial behavior of two-dimensional decagonal quasicrystal films with an adhesive layer due to thermal misfit

The interfacial behavior of a thin two-dimensional decagonal quasicrystal (QC) film bonded to a half plane with an adhesive layer is analyzed under thermal misfit. Under a perfect non-slipping contact condition at interface, a theoretical model is established based on the membrane assumption, with the governing integral–differential equations being constructed in terms of phonon interfacial shear stresses in one single and multiple films. Then, these governing equations are numerically solved by Chebyshev polynomials, obtaining the phonon interfacial shear and internal normal stresses as well as the longitudinal displacement in the QC film. Finally, the effects of neighboring films, material mismatch, adhesive layer, the geometry of QC film, and temperature variation are briefly discussed on the interfacial response. It is found that the adhesive layer plays as a softening medium which can greatly reduce the value of normal and shear stresses and weaken the singularity of shear stresses near the ends of films. These findings will be instructive to the accurate measurement and safety monitoring of QC films.

Novel results for separate families of fuzzy-dominated mappings satisfying advanced locally contractions in b-multiplicative metric spaces with applications

The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work.

Traction–separation law for bridged cracks at immiscible polymers interface

A traction–separation law for bonds at cracks bridged zones at immiscible polymers interface is developed. It is assumed that bonds at the crack bridged zone between different polymers are formed by bundles of the polymer molecules–promoter of adhesion. It is taken into account that molecules of the polymer-promoter can form one-stitch and many-stitch bonds between polymers. Nonlinear deformation curves of bonds at the crack bridged zone were obtained on this basis. The model of different materials joint with bridged interface crack is used to simulate immiscible polymers adhesive bonding. The system of singular integral-differential equations is applied for evaluation of bonds traction for the polymers interface bridged crack. The analysis of the bond traction–separation law parameters influence on the numerical iterative solution convergence of the integral-differential equations system has been performed. Evaluations of the polymer joint fracture toughness were performed using the non-local two-parameter criterion for quasi-static growth of bridged cracks.

Fuzzy dominated nonlinear operators with applications

The objective of this research is to propose some new fixed point theorems for fuzzy-dominated operators that satisfy a nonlinear contraction on a closed ball in a complete b-multiplicative metric space. Our strategy involves the use of a combination of two distinct kinds of mappings: one belongs to a weaker class of strictly increasing mappings, and the other is a class of dominated mappings. In order to demonstrate the validity of our new findings, we provide instances that are both illustrative and substantial. Finally, in order to illustrate the novelty of our findings, we provide applications that allow us to derive the common solution to integral and fractional differential equations. Our findings have a significant impact on the interpretation of a large number of previously published studies, both present and historical.

On the Thermally Induced Interfacial Behavior of Thin Two-Dimensional Hexagonal Quasicrystal Films with an Adhesive Layer

The mechanical response of a quasicrystal thin film is strongly affected by an adhesive layer along the interface. In this paper, a theoretical model is proposed to study a thin two-dimensional hexagonal quasicrystal film attached to a half-plane substrate with an adhesive layer, which undergoes a thermally induced deformation. A perfect non-slipping contact condition is assumed at the interface by adopting the membrane assumption. An analytical solution to the problem is obtained by constructing governing integral–differential equations for both single and multiple films in terms of interfacial shear stresses that are reduced to a linear algebraic system via the series expansion of Chebyshev polynomials. The solution is compared to that without adhesive layers, and the effects of the aspect ratio of films, material mismatch, and the adhesive layer, as well as the interaction between films, are discussed in detail. It is found that the adhesive layer can soften the localized stress concentration. This study is instructive to the accurate safety assessment and functional design of a quasicrystal film system.

An equivalent formulation of Sonine condition

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point.

A novel approach to solving system of integral partial differential equations based on hybrid modified block‐pulse functions

In nonlinear this paper, we present a new approach to solve system of two‐dimensionalFredholm–Volterra integral partial differential equations. This approach is based on a new hybrid of two‐dimensional Bernoulli polynomials and modified block‐pulse functions. Different from the traditional methods, in this new approach, the operational matrices are considered. Furthermore, by solving this matrix equation and using the collocation method, approximate solutions for the problem are obtained. Finally, error analysis and three practical examples are provided accuracy and efficiency illustrate our approach.

Lagrange formula conjugate third order differential equation

The paper considers a boundary value problem for a third order with no smooth coefficients and pure derivatives. Odds. This is due to the fact to introduce the concept of the conjugate Green's function. It is very difficult to write the form of the conjugate differential operator corresponding to equation in the Lagrange sense. Therefore, in this work, without using strict conditions smoothness under the conditions and boundedness, an explicit form is found conjugate operator since the initial-boundary value problem for integral-differential equations has been studied based on the introduction special conjugate systems in the form of an integral-algebraic equations’ system. In this article, it can be said that Green's function is considered based on Lagrange's formula for the third-order differential equation with boundary conditions and its conjugate.

Machine Learning Through Physics–Informed Neural Networks: Progress and Challenges

Physics-Informed Neural Networks (PINNs) represent a groundbreaking approach wherein neural networks (NNs) integrate model equations, such as Partial Differential Equations (PDEs), within their architecture. This innovation has become instrumental in solving diverse problem sets including PDEs, fractional equations, integral-differential equations, and stochastic PDEs. It's a versatile multi-task learning framework that tasks NNs with fitting observed data while simultaneously minimizing PDE residuals. This paper delves into the landscape of PINNs, aiming to delineate their inherent strengths and weaknesses. Beyond exploring the fundamental characteristics of these networks, this review endeavors to encompass a wider spectrum of collocation-based physics-informed neural networks, extending beyond the core PINN model. Variants like physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN) constitute pivotal aspects of this exploration. The study accentuates a predominant focus in research on tailoring PINNs through diverse strategies: adapting activation functions, refining gradient optimization techniques, innovating neural network structures, and enhancing loss function architectures. Despite the extensive applicability demonstrated by PINNs, surpassing classical numerical methods like Finite Element Method (FEM) in certain contexts, the review highlights ongoing opportunities for advancement. Notably, there are persisting theoretical challenges that demand resolution, ensuring the continued evolution and refinement of this revolutionary approach.

Fractional Partial and Integral Differential Equations and Novel Conformable Double (Laplace -Sumudu) Transform

Fractional Partial and Integral Differential Equations and Novel Conformable Double (Laplace -Sumudu) Transform

Physical and mathematical modeling of coding technology and RFID marking system

The increase in production volumes in the modern world leads to the need for the growth of various systems for encoding information about a particular product, moving on to the topic of improving labeling systems. This study examines the possibilities for further improvement of the RFID (Radio Frequency Identification) marking system, which is currently in greatest demand for classifying various units of goods. General information about this technology in a general sense is provided, after which, from the point of view of physical and mathematical modeling, the method of using electromagnetic induction to read data from markings is described in detail. During the technology analysis, various mathematical methods were used, including partial differential equations, integral equations and other physical and mathematical expressions for a general analysis of the processes under study. The conclusion provides general conclusions regarding the study, as well as information on the need for further improvement of this technology using various methods.

High-degree polynomial approximations for solving linear integral, integro-differential, and ordinary differential equations

High-degree polynomial approximations for solving linear integral, integro-differential, and ordinary differential equations

Boyd-Wong type contractions in generalized parametric bipolar metric space

In this manuscript, we introduce a new notion of generalized parametric bipolar metric space as a generalization of generalized parametric space and bipolar metric space. We also introduce Boyd-Wong type contractions for covariant and contravariant mappings to prove the fixed point results in the newly defined space. Some examples are also provided to illustrate the main results. Some corollaries are given of our results which shows the existence of fixed point for Banach type covariant and contravariant contractions. We solve integral and fractional differential equations with the help of proved results.