System Of Integral Equations Research Articles

A class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology

We study a class of free boundary problems describing the local-nonlocal cross-diffusion models with two species in ecology. The first is a native species with nonlocal cross-diffusion, and the second is an invasion species with local cross-diffusion. Since the local and nonlocal cross-diffusions are taken into account, the models are described by the strongly coupled systems of integral and differential equations. In addition, the position of each free boundary is determined by the Stefan condition. By using the transformations of function and variable, various estimates and contraction mapping theorem, we show the global existence, uniqueness and regularity of solutions for the free boundary problems. Besides, the long time behavior of the solution for a special type of local-nonlocal cross-diffusion competition model is also obtained.

On Sawada-Kotera and Kaup-Kuperschmidt integrable systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the M2 -extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.

PARASIAN OVER PARISIAN, HOW MUCH EARLIER SHOULD ONE EXERCISE?

In this paper, an integral equation approach used for pricing American-style Parisian options is extended to pricing Parasian options, after overcoming an additional difficulty of losing the “reset” feature of the latter in addition to still dealing with the high nonlinearity associated with American-style exercises. More specifically, compared with an American Parisian option, the accumulative feature of the “tracking clock” time results in a pair of coupled three-dimensional (3D) Partial Differential Equation (PDE) systems instead of coupled two-dimensional (2D) and 3D PDE systems. Upon successfully establishing the couple integral equation systems, we have successfully computed the option price and the optimal exercise price and compared the results with those of its Parisian counterpart. Through the comparison, we can directly observe how the accumulativeness of the “tracking clock” time affects the option and the optimal exercise prices and provide some meaningful financial explanations.

Reduced basis method for the elastic scattering by multiple shape-parametric open arcs in two dimensions

We consider the elastic scattering problem by multiple disjoint arcs or cracks in two spatial dimensions. A key aspect of our approach lies in the parametric description of each arc's shape, which is controlled by a potentially high-dimensional, possibly countably infinite, set of parameters. We are interested in the efficient approximation of the parameter-to-solution map employing model order reduction techniques, specifically the reduced basis method. Initially, we utilize boundary potentials to transform the boundary value problem, originally posed in an unbounded domain, into a system of boundary integral equations set on the parametrically defined open arcs. Our aim is to construct a rapid surrogate for solving this problem. To achieve this, we adopt the two-phase paradigm of the reduced basis method. In the offline phase, we compute solutions for this problem under the assumption of complete decoupling among arcs for various shapes. Leveraging these high-fidelity solutions and Proper Orthogonal Decomposition (POD), we construct a reduced-order basis tailored to the single arc problem. Subsequently, in the online phase, when computing solutions for the multiple arc problem with a new parametric input, we utilize the aforementioned basis for each individual arc. To expedite the offline phase, we employ a modified version of the Empirical Interpolation Method (EIM) to compute a precise and cost-effective affine representation of the interaction terms between arcs. Finally, we present a series of numerical experiments demonstrating the advantages of our proposed method in terms of both accuracy and computational efficiency.

Elastic Fields Around Multiple Stiff Prestressed Arcs Located on a Circle

Abstract The plane strain problem of an isotropic elastic matrix subjected to uniform far-field load and containing multiple stiff prestressed arcs located on the same circle is considered. The boundary conditions for the arcs are described by those of either Gurtin–Murdoch or Steigmann–Ogden theories in which the arcs are endowed with their own elastic energies. The material parameters for each arc can in general be different. The problem is reduced to the system of real variables hypersingular boundary integral equations in terms of two scalar unknowns expressed via the components of the stress tensors of the arcs. The unknowns are approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The coefficients in series are found from the system of linear algebraic equations that are solved using the collocation method. The expressions for the stress intensity factors are derived and numerical examples are presented to illustrate the influence of governing dimensionless parameters.

On the exploration of new solitary wave solutions for the classical integrable Kuralay-IIA system of equations

In this paper, we derive various new optical soliton solutions for the coupled Kuralay-IIA system of equations using an innovative solution approach known as the ϕ 6 − model expansion technique. This solution methodology employs a traveling wave transformation to reduce the considered problem into an easily solvable higher-order ordinary differential equation. Unlike other existing related methods, this solution approach adopted here allows us to extract a rich list of diverse exact soliton solutions for the considered problem. The obtained solutions incorporate the Jacobi elliptic functions which are shown to degenerate into trigonometric and hyperbolic function solutions. These solutions exhibit distinct wave structures consisting of dark, bright, rational, periodic, singular and mixed optical solitons profiles. In exploring the impact of spatial and temporal variables on the wave patterns of the considered model, physical structures of some of the obtained solitons solutions are characterized through 3D, contour and 2D wave profiles for selected parameter values. This not only ensures the validity of the solutions as well as the constraints arising from the solution technique but also offers researchers a deeper understanding of the properties of the considered problem. The outcomes here demonstrate the applicability, versatility and efficiency of the considered solution approach for deriving diverse new soliton solutions for even more complex systems of nonlinear evolution equations.

Investigating fractal fractional PDEs, electric circuits, and integral inclusions via (ψ,ϕ)-rational type contractions

The fixed point theory has been generalized mainly in two directions. One is the extension of the spaces, and the other is relaxing and generalizing the contractions. This paper aims to establish novel fixed point results of rational type generalized (ψ,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\psi ,\\phi )$$\\end{document}-contractions in the context of extended b-metric spaces. This will allow us to analyze generalized rational type contraction in a more relaxed and diversified framework in the light of the characteristics of (ψ,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\psi ,\\phi )$$\\end{document}. Some existing rational-type contractions have been recalled in this direction, and others are defined. New fixed point results have been established by utilizing the properties of ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi$$\\end{document} and ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi$$\\end{document} and applying the iteration technique. Moreover, the established results are employed to investigate the stability of fractal and fractional differential equations and electric circuits. For the reliability of the established results, examples and applications to the system of integral inclusions and system of integral equations are presented.

The Smirnov Property for Weighted Lebesgue Spaces

We establish lower norm bounds for multivariate functions within weighted Lebesgue spaces, characterised by a summation of functions whose components solve a system of nonlinear integral equations. This problem originates in portfolio selection theory, where these equations allow one to identify mean-variance optimal portfolios, composed of standard European options on several underlying assets. We elaborate on the Smirnov property—an integrability condition for the weights that guarantees the uniqueness of solutions to the system. Sufficient conditions on weights to satisfy this property are provided, and counterexamples are constructed, where either the Smirnov property does not hold or the uniqueness of solutions fails.

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered Ψ–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution.

Influence of specularity factor on heat transport in nanoribbons of different sizes

In this paper, the phonon heat transfer in two-dimensional conductors with different types of phonon reflection from the boundaries is examined. Heat conductors with arbitrary ratios of width W to length L are studied assuming that the mean free path between phonon-phonon collisions is infinite. The integral equations for the angular distribution functions of the incident and reflected phonons at a specific point on the conductor boundary were proposed. To solve these integral equations a new iterative method is proposed. The proposed iterative approach formally corresponds to taking into account subsequent collisions of phonons with the edges of the conductor. It ensures the convergence of the desired solution for any W/L ratio and for any value of the specularity coefficient p. Interpolation formulae are found to describe with sufficient accuracy the solution of the system of integral equations in the entire region of the specularity coefficient p from 0 to 1, and the W/L ratios ranging from 10 to 0.01. These formulae allow the construction of the isolines of the thermal conductance coefficient values, from which it is possible to determine the necessary values of W/L and parameter p to get the desired value of the thermal conductance.

Homotopy Perturbation Method for Semi-Bounded Solution of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this paper, we introduce the standard Homotopy Perturbation Method (HPM) for obtaining semi-bounded solutions of the first kind of system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. We use the Gauss elimination technique to reduce the system of CSIEs to a diagonal triangle system of algebraic equations. We then apply the HPM to solve the resulting equations. By applying the theory of semi-bounded solutions of CSIEs, we can determine the inverse operators for the first kind of CSIEs. We demonstrate that the proposed method is exact for the system of characteristic SIEs, regardless of the choice of initial guesses (in the Holder class of functions). To illustrate the validity and accuracy of our proposed method, we supply and analyse three examples. We compare the results obtained using our method with those obtained using the Chebyshev collocation and Galerkin methods. Our method includes the ability to solve the complex-valued system of CSIEs. Based on the numerical results, we conclude that the HPM is more dominant than the other methods.

Superconvergent scheme for a system of green Fredholm integral equations

In this study, a numerical scheme to a system of second-kind linear Fredholm integral equations featuring a Green's kernel function is proposed. This involves introducing Galerkin and iterated Galerkin (IG) methods based on piecewise polynomials to tackle the integral model. A thorough analysis of convergence and error for these proposed methods is carried out. Firstly, the existence and uniqueness of solutions for the Galerkin and iterated Galerkin methods are established. Later, the order of convergence is derived using tools from functional analysis and the boundedness property of Green's kernel. The Galerkin scheme has O(hα) order of convergence. Next, the superconvergence of the iterated Galerkin (IG) method is established. The IG method exhibits O(hα+α⁎) order of convergence. Theoretical findings are validated through extensive numerical experiments.

Solving system of nonlinear Abel’s integral equations in frequency domain

In this paper, we solve both linear and nonlinear systems of Abel’s integral equations by combining the Laplace transform with Bernstein polynomials. In the proposed approach, the unknown functions are approximated using Bernstein polynomials. The Laplace transform is then applied to convert the equations from the time domain to the frequency domain. This results in a nonlinear system in terms of unknown coefficients, which is solved using Newtons iterative method. The unknown function is thus approximated. To illustrate the efficacy of the proposed method, we present detailed analytical and numerical examples.

Tripled Fixed Point Methodologies for Finding a Solution to a System of Nonlinear Integral Equations on Orthogonal Metric Spaces

Tripled Fixed Point Methodologies for Finding a Solution to a System of Nonlinear Integral Equations on Orthogonal Metric Spaces

An efficient recursive technique with Padé approximation for a kind of Lane–Emden type equations emerging in various physical phenomena

The study numerically examined a class of nonlinear singular differential problems known as the Lane–Emden differential equation, which emerges in numerous real-world situations. The primary goal of this work is to formulate a computationally efficient iterative technique for solving the nonlinear Lane–Emden initial value problems. The proposed approach is a hybrid of the homotopy perturbation method and the Padé approximation. The nonlinear singular Lane–Emden initial value problem (SLEIVP) is transformed into an equivalent recursive integral employing the Picard’s approach. To resolve the singularity and nonlinearity, the recursive integral equation is transformed into a system of integral equations by using the homotopy notion. Furthermore, to enhance the convergence rate of the technique, Padé approximation is taken into account. The convergence analysis for the proposed approach is also conducted. The present technique is tested on SLEIVPs and numerical findings are compared with the existing techniques, to demonstrate the accuracy, effectiveness and ease of use.

Common Fixed Point Theorems in multiplicative $\mathfrak{m}$-metric space with Applications to the System of Multiplicative Integral Equations and Numerical Results

This paper presents common fixed point results for a pair of self-mappings in multiplicative $\mathfrak{m}$-metric space. Also, we present the multiplicative partial metric structure as a specific case of a multiplicative m-metric space and demonstrate some common fixed point results. To support our conclusions, we present an illustrative example with discontinuous self-mappings. We also provide numerical iterations to approximate the common fixed point and graphs to visually substantiate our findings. As the consequences of our results, we demonstrate several common fixed point results in $m$-metric space and partial metric space. Our findings generalize various fixed point results from the literature. Furthermore, we employ the results to demonstrate the existence and uniqueness of solutions to a system of multiplicative Fredholm integral equations.

Analytical solution of two non‐identical edge cracks in an infinite strip under anti‐plane shear wave

AbstractThis article presents an extensive analytical solution addressing the interaction between two non‐identical edge cracks in an infinite orthotropic strip under anti‐plane shear waves. Most studies assume identical cracks or single edge crack in a strip, but this research breaks new ground by considering cracks of different sizes. By incorporating mixed‐type boundary conditions, the study derives dual integral equations. These equations are then transformed into a singular integral equation of Cauchy type with the aid of a trial solution and contour integration technique. The singular integral equation is further converted into a system of integral equations, which are solved numerically utilizing Jacobi polynomials. The obtained solutions are utilized to derive expressions for the stress intensity factor (SIF) and crack opening displacement (COD) at the crack tip using Krenk's interpolation formulae. The derived results are presented graphically and compared against existing solutions for single edge crack and symmetric edge cracks in static scenario.

Solvability analysis of the nonlinear integral equations system arising in the logistic dynamics model in the piecewise constant kernels case

Solvability analysis of the nonlinear integral equations system arising in the logistic dynamics model in the piecewise constant kernels case

Longitudinal Shear of a Composite Material in Which the Binder and Inclusions are Weakened by Cohesive Cracks

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth. Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained. Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined. Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations". Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1. Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes. Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.