Functional Integral Equations Research Articles

Adomian Decomposition Method and Block by Block Method for Solving Nonlinear Functional Volterra Integral Equation in Two-dimensions

This work proposes a new defintion of the nonlinear functional Volterra integral equation ib 2D of the second kind with continuous kernel (NT-DFVIE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions are obtained by two powerful methods Adomian Decomposition Method (ADM) and Block by block method (BBM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

Investigation of conditions for preserving global solvability of operator equations by means of comparison systems in the form of functional-integral equations in $\mathbf{C}[0;T

Let $U$ be the set of admissible controls, $T>0$ and it be given a scale of Banach spaces $W[0;\tau]$, $\tau\in(0;T]$, such that the set of constrictions of functions from $W=W[0;T]$ to a closed segment $[0;\tau]$ coincides with $W[0;\tau]$; $F[\cdot;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. For the operator equation $x=F[x;u]$, $x\in W$, we introduce a comparison system in the form of functional-integral equation in the space $\mathbf{C}[0;T]$. We establish that, under some natural hypotheses on the operator $F$, the preservation of the global solvability of the comparison system pointed above is sufficient to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation. This fact itself is analogous to some results which were obtained by the author earlier. The central result of the paper consists in a set of signs for stable global solvability of functional-integral equations mentioned above which do not use hypotheses similar to local Lipschitz continuity of the right-hand side. As a pithy example of special interest, we consider a nonlinear nonstationary Navier–Stokes system in the space $\mathbb{R}^3$.

New Results on Ulam Stabilities of Nonlinear Integral Equations

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results.

On refined rational convex contractions with applications to matrix and implicit functional integral equations

On refined rational convex contractions with applications to matrix and implicit functional integral equations

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods.

A constrained problem of a nonlinear functional integral equation subject to the pantograph problem

A constrained problem of a nonlinear functional integral equation subject to the pantograph problem

Study of numerical treatment of functional first-kind Volterra integral equations

<abstract><p>First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.</p></abstract>

Solving hybrid functional-fractional equations originating in biological population dynamics with an effect on infectious diseases

<abstract><p>This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory.</p></abstract>

NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR A SECOND ORDER IMPULSIVE SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED MAXIMA

A two-point nonlinear boundary value problem for a second order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. By applying some transformations is obtained a system of nonlinear functional integral equations. The existence and uniqueness of the solution of the nonperiodic two-point boundary value problem are reduced to the one valued solvability of the system of nonlinear functional integral equations in Banach space . The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional integral equations.

Oscillatory Solution of a Convolutional Volterra Integral Equation

Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton’s method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second objective of the paper is to extend and improve some earlier results to nonlinear HTFIEs. The authors prove two new theorems with regard to the existence and uniqueness of solutions (EUSs) of HTFIEs via a technique called progressive contractions, which belongs to T. A. Burton, and the Chebyshev norm and complete metric.

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

Theoretical and numerical analysis of a first-kind linear Volterra functional integral equation with weakly singular kernel and vanishing delay

Frequency domain Green’s function and boundary integral equations for multifield materials and quasicrystals

Present paper considers anisotropic elastic, magnetoelectroelastic and quasicrystal solids. It is shown that the equations of time-harmonic motion and constitutive relations for considered materials can be presented in the compact and unified form. The matrix approach is proposed for derivation of 3D time-harmonic Green’s functions for considered materials. The cases of each material type are studied separately. Two models of phason field dynamics in quasicrystals are considered in details. It is shown that in the case of phason field models based on the use of hydrodynamic formulations, phonon oscillations have an elastic nature, while the phason field leads to their decay due to phonon-phason interaction. The strict proof is given to the statement that the eigenvalues of the time-harmonic magnetoelectroelaticity problem are all positive. The paper also shows the application of the obtained time-harmonic Green’s functions in obtaining the boundary integral equations for the considered classes of problems. For this purpose, the novel approach is proposed, which utilizes only the symmetry of the unified material property tensor and the inertia matrix. Derived boundary integral equations can be used in the solution of boundary value problems for considered classes of materials.

The linear Fredholm integral equations with functionals and parameters

The theory of linear Fredholm integral-functional equations of the second kind with linear functionals and a parameter is considered. The necessary and sufficient conditions are obtained for the coefficients of the equation and those parameter values in the neighbourhood of which the equation has solutions. The leading terms of the asymptotics of the solutions are constructed. The constructive method is proposed for constructing a solution both in the regular case and in the irregular one. In the regular case, the solution is constructed as a Taylor series in powers of the parameter. In the irregular case, the solution is constructed as a Laurent series in powers of the parameter. The example is used to illustrate the proposed constructive theory and method.

An Investigation of a Nonlinear Delay Functional Equation with a Quadratic Functional Integral Constraint

This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L1[0,T] and, secondly, the existence of nonincreasing solutions in unbounded interval L1(R+). Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ∈L1[0,T] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ∈L1(R+) on some parameters. Finally, some examples are provided to verify our investigation.

Existence, uniqueness, and numerical approximation of solutions of a nonlinear functional integral equation

A general nonlinear functional integral equation has been studied. The considered equation is of Fredholm type and includes some well-known integral or functional equations, as special cases. By assuming some smoothness conditions on the functions in the equation, we prove the existence and uniqueness of the solution by means of a fixed point method. Then, we apply the so-called Nyström method in order to approximate the solution, which leads to a system of nonlinear algebraic equations. In the next step, the Picard iteration method is utilized for approximating the solution of the resulting system of nonlinear algebraic equations. The convergence of the numerical scheme is proved, its computational complexity is computed and finally some illustrative examples are included to demonstrate the efficiency and applicability of the technique.

Asymptotic Stability and Dependency of a Class of Hybrid Functional Integral Equations

Asymptotic Stability and Dependency of a Class of Hybrid Functional Integral Equations

New Aspects on the Solvability of a Multidimensional Functional Integral Equation with Multivalued Feedback Control

The current study demonstrates the existence of solutions to a multidimensional functional integral equation with multivalued feedback. We seek solutions for the multidimensional functional problem that is defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in BC(R+). Also, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions to that multidimensional functional equation. Additionally, we give an example and some particular cases to illustrate our outcomes.

Numerical solution of delay Volterra functional integral equations with variable bounds

This work presents a new numerical method for solving Volterra-type integro functional equations with variable bounds and mixed delay. This paper applies discrete orthogonal Hahn polynomials and their properties numerically. A discrete scalar product is associated with discrete orthogonal polynomials. Several numerical experiments (including linear and nonlinear) for multiple test problems are provided to validate the accuracy of this method.

On the solvability of a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator

In this paper, utilizing the technique of generalized Darbo's fixed‐point theorem associated with measure of noncompactness in Banach space, we analyze the existence of solution for a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator. The existing result was obtained to strengthen the ones mentioned previously in the literature. An example for a class of nonlinear functional integral equations is also presented to validate our main result. Finally, we propose the numerical method formed by the modified homotopy perturbation approach to resolving the problem with acceptable accuracy.