Volterra-Hammerstein Integral Equations Research Articles

Numerical Solution of Volterra-Hammerstein Integral Equation of the First Kind by Finite Difference Method Decomposition with Nyström Method

Numerical Solution of Volterra-Hammerstein Integral Equation of the First Kind by Finite Difference Method Decomposition with Nyström Method

Implicitly linear Jacobi spectral-collocation methods for two-dimensional weakly singular Volterra-Hammerstein integral equations

Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral-collocation method to solve two-dimensional weakly singular Volterra-Hammerstein integral equations based on smoothing transformation and implicitly linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the nonlinear Hammerstein term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. The weakly singular integral term is discretized by Jacobi Gauss quadrature formula which can absorb the weakly singular kernel function into the quadrature weight function and eliminate the influence of the weakly singular kernel on the method. Convergence analysis in the L∞-norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples.

Approximated superconvergent methods for Volterra Hammerstein integral equations

In this article, we apply Kumar–Sloan based Galerkin and multi-Galerkin methods for solving the nonlinear Volterra–Hammerstein integral equations with both smooth and weakly singular kernels. For smooth kernels, we use piecewise polynomials basis functions based on uniform mesh, whereas for weakly singular kernels, we use piecewise polynomials basis functions based on graded mesh, to derive the superconvergence results. We obtain the improved superconvergence rates using Kumar–Sloan based Galerkin and multi-Galerkin methods without going to the iterated Galerkin and iterated multi-Galerkin methods. Infact without going to the iterated versions, we obtain the superconvergence rates in this method as in the case of iterated Galerkin and iterated multi-Galerkin methods. Numerical examples are given to verify the theoretical results.

Solving Integral Equations via Fixed Point Results Involving Rational-Type Inequalities

In this study, we establish unique and common fixed point results in the context of a complete complex-valued b-metric space using rational-type inequalities. The presented work generalizes some well-known results from the existing literature. Furthermore, to ensure the validity of the findings, we have included some examples and a section on the existence of solutions for the systems of Volterra–Hammerstein integral equations and Urysohn integral equations, respectively.

The series expansions and blow-up time estimation for the solutions of convolution Volterra-Hammerstein integral equations

The series expansions and blow-up time estimation for the solutions of convolution Volterra-Hammerstein integral equations

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors.

Projection-iterated method for solving numerically the nonlinear mixed integral equation in position and time

In this work, we consider a nonlinear Volterra-Hammerstein integral equation of the second kind (V-HIESK). The existence of a unique solution to the integral equation, under certain conditions, is guaranteed. The projection method was used, as the best numerical method. This method is better than the previous methods in the possibility of obtaining the lowest relative error. The method of obtaining the best approximate solution, through the projection method, depends on presenting three consecutive algorithms and depends mainly on the method of iterative projection (P-IM). In each algorithm, we get the approximate solution and the corresponding relative error. Moreover, we demonstrated that the estimated error of P-IM in the first algorithm is better than that of the successive approximation method (SAM). Some numerical results were calculated, and the error estimate was computed, in each case. Finally, the numerical results have been compared between this method and previous research, and it is clear through the same examples that the relative error of the first algorithm is better than the comparative error in the other methods.

On generalized $ \Im_b $-contractions and related applications

<abstract><p>In this manuscript, using the idea of $ b $-simulation functions, certain common fixed point results via generalized $ \Im_b $-contractions are investigated in the context of b-metric spaces. These findings generalize and supplement various numbers of established results from the existing literature. Examples and applications are also provided for the authenticity of the presented work. Besides, we ensure the existence of a unique common solution for systems of Volterra-Hammerstein integral and Urysohn integral equations, respectively, by applying the established results.</p></abstract>

On solvability of Volterra-Hammerstein integral equations in two variables coordinatewise converging at infinity

On solvability of Volterra-Hammerstein integral equations in two variables coordinatewise converging at infinity

A numerical method for the solution of nonlinear Volterra Hammerstein integral equations of the third-kind

In this paper we analyze spectral collocation method on a special class of nonlinear Volterra Hammerstein integral equations of the third-kind. After explaining the method, convergence analysis is discussed and then some examples are considered to show the validity of the theoretical results.

On a Functional Integral Equation

In this paper, we establish some results for a Volterra–Hammerstein integral equation with modified arguments: existence and uniqueness, integral inequalities, monotony and Ulam-Hyers-Rassias stability. We emphasize that many problems from the domain of symmetry are modeled by differential and integral equations and those are approached in the stability point of view. In the literature, Fredholm, Volterra and Hammerstein integrals equations with symmetric kernels are studied. Our results can be applied as particular cases to these equations.

Projection and multi-projection methods for second kind Volterra-Hammerstein integral equation

In this article, we discuss the piecewise polynomial based Galerkin method to approximate the solutions of second kind Volterra-Hammerstein integral equations. We discuss the convergence of the approximate solutions to the exact solutions and obtain the orders of convergence $mathcal O(h^{r})$ and $mathcal O(h^{2r}),$ respectively, for Galerkin and its iterated Galerkin methods in uniform norm, where $h, ~r$ denotes the norm of the partition and smoothness of the kernel, respectively. We also obtain the superconvergence results for multi-Galerkin and iterated multi-Galerkin methods. We show that iterated multi-Galerkin method has the order of convergence $mathcal O(h^{3r})$ in the uniform norm. Numerical results are provided to demonstrate the theoretical results.

A high order approach for nonlinear Volterra-Hammerstein integral equations

<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>

Asymptotic Approximations for Solutions of Sturm–Liouville Differential Equations with a Large Complex Parameter and Nonsmooth Coefficients

This article deals with fundamental solutions of the Sturm–Liouville differential equations $$\left( P\left( x\right) y^{\prime }\right) ^{\prime }+\left( R\left( x\right) -\xi ^{2}Q\left( x\right) \right) y=0$$ with a complex parameter $$\xi $$ , $$\left\vert \xi \right\vert \gg 1$$ , whose coefficients $$P\left( x\right) $$ and $$Q\left( x\right) $$ are positive piecewise continuous functions while $$\left( P\left( x\right) Q\left( x\right) \right) ^{\prime }$$ and $$R\left( x\right) $$ belong to $$L^{p}$$ , $$p>1$$ . Two methods are suggested to reduce the problem to special Volterra or Volterra–Hammerstein integral equations with “small” integral operators, which can be treated by iterations that constitute asymptotic scales or by direct numerical methods. Asymptotic analysis of the iterations permits constructing uniform asymptotic approximations for fundamental solutions and their derivatives. A uniform with respect to integration limits analog of Watson’s lemma for finite range Laplace integrals is proved in this connection. When the coefficients belong to $$C^{N}$$ on intervals of continuity, a new simple direct method is suggested in order to derive explicit uniform asymptotic approximations of fundamental solutions as $$\operatorname{Re} \left( \xi \right) \gg 1$$ by a certain recurrent procedure. This method is expected to be more efficient than the WKB method. Theorems estimating errors of such approximations are given. Efficacy of the suggested uniform approximations is demonstrated by computing approximations for a combination of Bessel functions of large indices and arguments.

Neural field equations with neuron‐dependent Heaviside‐type activation function and spatial‐dependent delay

We introduce a neural field equation with a neuron‐dependent Heaviside‐type activation function and spatial‐dependent delay. The basic object of the study is represented by a Volterra–Hammerstein integral equation involving a discontinuous nonlinearity with respect to the state variable that is both time and space dependent. We replace the discontinuous nonlinearity by its multivalued convexification and obtain the corresponding Volterra–Hammerstein integral inclusion. We investigate the solvability of this inclusion using the properties of upper semicontinuous multivalued mappings with convex closed values. Based on these results, we study the solvability of an initial‐prehistory problem for the former neural field equation with the Heaviside‐type activation function. The application of multivalued analysis techniques allowed us to avoid some restrictive assumptions standardly used in the investigations of the solutions to neural field equations involving Heaviside‐type activation functions.

Galerkin and multi-Galerkin methods for weakly singular Volterra–Hammerstein integral equations and their convergence analysis

In this paper, we consider the Galerkin and iterated Galerkin methods to approximate the solution of the second kind nonlinear Volterra integral equations of Hammerstein type with the weakly singular kernel of algebraic type, using piecewise polynomial basis functions based on graded mesh. We prove that the Galerkin method converges with the order of convergence $${\mathcal {O}}(n^{-m})$$, whereas iterated Galerkin method converges with the order $${\mathcal {O}}(n^{-2m})$$ in uniform norm. We also prove that in iterated multi-Galerkin method the order of convergence is $${\mathcal {O}}(n^{-3m}) $$. Numerical results are provided to justify the theoretical results.

Solutions of Hammerstein equations in the space (Λ_1,Λ_2)BV(I^b_a)

In this paper, we study the form of Hammerstein integral equations $u(x)=v(x)+\lambda\int_{I^{b}_{a}}k(x,y)f(y,u(y))dy,(\lambda\in\mathbb{R})$ and Volterra Hammerstein integral equations in the condition of two-variables. Show the definition of $(\Lambda_{1},\Lambda_{2})$ bounded variation, write as $(\Lambda_{1},\Lambda_{2})BV(I^{b}_{a})$. If $v$, $k$ are $(\Lambda^{(1)},\Lambda^{(2)})BV(I^{b}_{a};\mathbb{R})$ functions and $f$ is a locally Lipschitz function, there exists a number $\rho>0$ such that when $|\lambda|<\rho$, Hammerstein integral equations has a unique solution. Give the proof and extend.

An iterative numerical method to solve nonlinear fuzzy Volterra-Hammerstein integral equations

An iterative numerical method to solve nonlinear fuzzy Volterra-Hammerstein integral equations

Spreading speeds of rabies with territorial and diffusing rabid foxes

A mathematical model is formulated for the fox rabies epidemic that swept through large areas of Europe during parts of the last century. Differently from other models, both territorial and diffusing rabid foxes are included, which leads to a system of partial differential, functional differential and differential-integral equations. The system is reduced to a scalar Volterra-Hammerstein integral equation to which the theory of spreading speeds pioneered by Aronson and Weinberger is applied. The spreading speed is given by an implicit formula which involves the space-time Laplace transform of the integral kernel. This formula can be exploited to find the dependence of the spreading speed on the model ingredients, in particular on those describing the interplay between diffusing and territorial rabid foxes and on the distribution of the latent period.

Optimization of Functional Integral Equations: A Spectral Approach

The spectral method of Elnagar and Kazemi (J. Comp. Appl. Math. 76(1–2):147–158, 1996), which yields spectral convergence rate for the approximate solutions of Volterra-Hammerstein integral equations, is generalized in order to solve the larger class of functional integral equation control systems with spectral accuracy. The proposed method is based on the idea of relating spectrally constructed grid points to the structure of projection operators. These operators will be used to approximate the control vector and the associated state vector. Numerical examples are included to demonstrate the accuracy of the proposed method.