Type Integral Equations Research Articles

PULSE BENDED BEAM WITH BINARY BOUNDARY CONDITIONS

A theoretical study of the movement of a beam with binary boundary conditions at a short-term pulsed load is carried out in the article. The non-stationary oscillations of the beam, caused by short-term force loading, are evenly distributed along its length. Beam movement is considered in two stages. It is assumed that the conditions for fixing the ends of the beam are associated with the direction (sign) of their rotation when the beam is bending. It is assumed that under the action of the load, the edges of the beam are rigidly clamped, that is, the angles of their rotation are equal to zero, and after unloading and changing the sign of the curvature of the beam they become pivotally supported in cylindrical hinges. According to the second boundary conditions, the free oscillation of the beam passes. Due to the use of the method of compensating loads, the analytical part of the isolation is expressed in rows of sines. Since the system has an asymmetric elastic characteristic with respect to the position of static equilibrium, the method of adding solutions is used to achieve this goal. As a result, the calculation is reduced to the numerical solution of the Volterra type integral equation on a computer. In addition, an approximation is proposed, which gives simpler calculation formulas than a piecewise linear approximation. The proposed approximation allows us to calculate the value of the boundary moments and obtain the formula for the beam deflections. Numerical analysis showed that for certain durations of force loading, the beam deflection amplitude with pinched edges in the direction of the external impulse is less than the deflection amplitude in the opposite direction during the movement of the unloaded beam with hinged edges. This dynamic effect is characteristic of systems with an asymmetric characteristic of elasticity, which occurs when a binary fixing of the edges of the beam or when it is supported by unilateral elastic supports. Formulas are derived by which it is possible to calculate the time when the effect of asymmetry will be most clearly expressed, and also in which case it will be absent. Numerical results confirming the adequacy of the proposed dependencies are presented.

The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform

In this paper, the asymptotic expansions for the solution about zero and infinity are formulated via Laplace transform for linear Volterra integral equation with weakly singular convolution kernel. The expansions about zero and infinity, as well as their Pade approximations, are used to approximate the solution when the argument is small and large, respectively, and the Pade approximations are more accurate. The methods are also valid to solve some other Volterra type integral equations including linear Volterra integro-differential equations, fractional integro-differential equations, and system of singular Volterra integral equations of the second kind with convolution kernels. Some examples confirm the correctness of the methods and the effectiveness of the asymptotic expansions. They show that numerical methods are only necessary in a small interval in practical computation when uniform high precision evaluations are needed for solving these kinds of Volterra integral equations.

A new recursive formulation of the Tau method for solving linear Abel–Volterra integral equations and its application to fractional differential equations

In this paper, the recursive approach of the Tau method is developed for numerical solution of Abel–Volterra type integral equations. Due to the singular behavior of solutions of these equations, the existing spectral approaches suffer from low accuracy. To overcome this drawback we use Muntz–Legendre polynomials as basis functions which have remarkable approximation to functions with singular behavior at origin and express Tau approximation of the exact solution based on a sequence of basis canonical polynomials that is generated by a simple recursive formula. We also provide a convergence analysis for the proposed method and obtain an exponential rate of convergence regardless of singularity behavior of the exact solution. Some examples are given to demonstrate the effectiveness of the proposed method. The results are compared with those obtained by existing numerical methods, thereby confirming the superiority of our scheme. The paper is closed by providing application of this method to approximate solution of a linear fractional integro-differential equation.

Dirichlet problem on the half-line for a forced Burgers equation with time-variable coefficients and exactly solvable models

We consider a forced Burgers equation with time-variable coefficients and solve the initial-boundary value problem on the half-line 0 < x < ∞ with inhomogeneous Dirichlet boundary condition imposed at x=0. Solution of this problem is obtained in terms of a corresponding second order ordinary differential equation and a second kind singular Volterra type integral equation. As an application of the general results, we introduce three different Burgers type models with specific damping, diffusion and forcing coefficients and construct classes of exactly solvable models. The Burgers problems with smooth time-dependent boundary data and an initial profile with pole type singularity have exact solutions with moving singularity. For each model we provide the solutions explicitly and describe the dynamical properties of the singularities depending on the time-variable coefficients and the given initial and boundary data.

Exact and asymptotic bending analysis of microbeams under different boundary conditions using stress‐derived nonlocal integral model

AbstractEringen's nonlocal differential elasticity is widely applied to model nano‐ and micro‐structures, although previous studies have shown some inconsistences, e.g., the nonlocal parameter maybe has no effect on the deflection of Euler‐Bernoulli and Timoshenko beams subjected to some kinds of boundary and loading conditions. In this paper, the static bending analysis of Euler‐Bernoulli and Timoshenko beams is performed with the application of stress‐driven nonlocal integral model. The Fredholm type integral constitutive equations of the first kind are transformed to Volterra integral equations of the first kind through simply adjusting the limit of integrals, and the general solutions to the deflection as well as bending moment and so on are derived and obtained explicitly through solving the integro‐differential governing equations with the Laplace transformation. Exact solutions are derived explicitly for different loading and boundary conditions, especially for the paradoxical beam problem while using nonlocal differential model. The analytical and asymptotic expressions of the beam deflections obtained in this paper are validated against to those existing analytical and numerical results in literature. It is clearly established that, with the application of the stress‐driven nonlocal integral model, a consistent toughening effect can be obtained for both Euler‐Bernoulli and Timoshenko beams.

Long time behavior of a mean-field model of interacting neurons

We study the long time behavior of the solution to some McKean–Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant probability measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant probability measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean–Vlasov equation.

Multidimensional inhomogeneous mixed initial-boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous mixed initial-boundary value problem for the nonlinear multidimensional Schrödinger equation, formulated on upper right-quarter plane. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. Also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions. Our approach to get well-posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument. To get a linear theory for the multidimensional model we propose general method based on Laplace approach and theory Cauchy type integral equations. To get smooth solutions in L∞ we modify a method based on the factorization for the free Schrödinger evolution group. The advantage of our approach is that it can also be applied to non-integrable equations and arbitrary boundary conditions. This approach is new and it is not standard. We believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis.

General 2 × 2 system of nonlinear integral equations and its approximate solution

In this note, we consider a general 2 × 2 system of nonlinear Volterra type integral equations. The modified Newton method (modified NM) is used to reduce the nonlinear problems into 2 × 2 linear system of algebraic integral equations of Volterra type. The latter equation is solved by discretization method. Nystrom method with Gauss–Legendre quadrature is applied for the kernel integrals and Newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points. Existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained. Finally, numerical examples are provided to show the validity and efficiency of the method presented. Numerical results reveal that the proposed methods is efficient and accurate. Comparisons with other methods for the same problem are also presented.

Projection and multi projection methods for nonlinear integral equations on the half-line

In this article, Galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear Hammerstein type integral equation on the half-line for both convolution and non-convolution kernels using the space of piecewise polynomial subspace. We prove that the approximate and iterated approximate solution in Galerkin and collocation methods converges with the order O ( n − r ) and O ( n − 2 r ) respectively in the infinity norm, where n − 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation. We improve these results further by considering multi-projection methods. In fact we show that iterated approximate solution in multi-Galerkin, multi-collocation converges with the order O ( n − 4 r ) . Numerical results are presented to confirm the theoretical results.

A Necessary Condition for Certain Integral Equations with Negative Exponents

This paper is devoted to studying the existence of positive solutions for the following integral system \(\left\{ {\begin{array}{*{20}{c}} {u\left( x \right) = \int_{{\mathbb{R}^n}} {{{\left| {x - y} \right|}^\lambda }{v^{ - q}}\left( y \right)dy,} } \\ {v\left( x \right) = \int_{{\mathbb{R}^n}} {{{\left| {x - y} \right|}^\lambda }{u^{ - p}}\left( y \right)dy,} } \end{array}} \right.p,q > 0,\lambda \in \left( {0,\infty } \right),n \geqslant 1\). It is shown that if (u, v) is a pair of positive Lebesgue measurable solutions of this integral system, then \(\frac{1}{{p - 1}} + \frac{1}{{q - 1}} = \frac{\lambda }{n}\), which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations.

Modeling the Age-Dependent Infectiousness of Diseases: An Integral Equation Approach

Many mathematical models developed through differential equations to describe the age dependent infectiousness of diseases, face the complexity of modelling heterogenic behavior of transmission. There, many of the cases assume the host to stay in the same risk class regardless of the age of the hosts. The proposed model mimics the infectiousness according to the age-scale of an individual via integral equation approach. This model indicates the applicability of Fredholm type integral equations with degenerated kernel. Introducing biological, behavioral and environmental influences provokes to address the accumulating nature of different factors in modelling the risk of getting infected. The risk of getting infected is modeled by the inability of responding with acquired immunity and the accumulated risk given from the other individuals in each age group via the mobility patterns. Within this approach environmental stimulus are modeled via periodic functions in order to describe the stochastic behavior of the spreading capabilities. In this study, the behavioral analysis evaluates the maximum risk of getting infectious in the considered parsimonious approach. And the sensitivity analysis describes the contribution of the mobility risk and stochastic nature on the overall risk. Further the model guides to formulate hypotheses and data collection strategies to measure the risk of a disease.

Some Fixed Point Theorems in Menger Probabilistic Partial Metric Spaces with Application to Volterra Type Integral Equation

In this paper, we introduce the notion of Menger probabilistic partial metric space and prove some fixed point theorems in the framework of such spaces. Some examples and an application to Volterra type integral equation are given to support the obtained results. Finally, we apply successive approximations method to find a solution for a Volterra type integral equation with high accuracy.

Numerical solution of the age structure optimization problem for basic types of power plants

Our paper addresses an integral model of the large electric power system optimal development. The model takes into account the age structure of the main equipment, which is divided into several types regarding its technical characteristics. This mathematical model is a system of Volterra type integral equations with variable integration limits. The system describes the balance between the given demand for electricity, the commissioning of new equipment and the dismantling of obsolete equipment, as well as the shares of different types of power plants in the total composition of the electric power system equipment. Based on the developed model, we got numerical solution to the problem of finding the optimal strategy for replacing equipment with a minimum of the cost functional. The case study is the Unified Electric Power System of Russia. Calculations of the forecast for development of the electric power system of Russia until 2050 were made using real-life data.

A family of measures of noncompactness in the spaceLploc(ℝN) and its application to some nonlinear convolution type integral equations

AbstractThe aim of the present paper is to introduce a new family of measures of noncompactness on the Frechet space LPloc(ℝN) (1 ≤ p < ∞). Further, we prove a fixed point theorem on the family of ...

A nonstandard Volterra integral equation on time scales

Abstract This paper introduces the more general result on existence, uniqueness and boundedness for solutions of nonstandard Volterra type integral equation on an arbitrary time scales. We use Lipschitz type function and the Banach’s fixed point theorem at functional space endowed with a suitable Bielecki type norm. Furthermore, it allows to get new sufficient conditions for boundedness and continuous dependence of solutions.

Some New Applications of Elzaki Transform for Solution of Linear Volterra Type Integral Equations

Volterra type integral equations have diverse applications in scientific and other fields. Modelling physical phenomena by employing integral equations is not a new concept. Similarly, extensive research is underway to find accurate and efficient solution methods for integral equations. Some of noteworthy methods include Adomian Decomposition Method (ADM), Variational Iteration Method (VIM), Method of Successive Approximation (MSA), Galerkin method, Laplace transform method, etc. This research is focused on demonstrating Elzaki transform application for solution of linear Volterra integral equations which include convolution type equations as well as one system of equations. The selected problems are available in literature and have been solved using various analytical, semi-analytical and numerical techniques. Results obtained after application of Elzaki transform have been compared with solutions obtained through other prominent semi-analytic methods i.e. ADM and MSA (limited to first four iterations). The results substantiate that Elzaki transform method is not only a compatible alternate approach to other analytic methods like Laplace transform method but also simple in application once compared with methods ADM and MSA.

On the approximation of Volterra integral equations with highly oscillatory Bessel kernels via Laplace transform and quadrature

The present work focuses on formulating a numerical scheme for approximation of Volterra integral equations with highly oscillatory Bessel kernels. The application of Laplace transform reduces integral equations into algebraic equations. By application of inverse Laplace transform solution is presented as an integral along a smooth curve extending into the left half of the complex plane, which is then evaluated by quadrature. Some model problems are solved and the results are compared with other available methods. The supremacy of the present method is that the transformed problem becomes non oscillatory. Consequently such types of integral equations with highly oscillatory kernels can be approximated very effectively with large values of oscillation parameter. A small amount of work such as Clenshaw-Curtis-Filon type methods are available for efficient approximation of highly oscillatory integral equations.

Water wave interaction with dual asymmetric non-uniform permeable plates using integral equations

In this study, we analyse the effect of two submerged unequal permeable plates in the propagation of water waves under the assumptions of linear water wave theory. The permeability of the plates varies along the depth of submergence of the plates. The plates are submerged in water of uniform depth. The velocity potential is expanded by using Havelock’s expansion of water wave potential. The associated boundary value problem is transformed into two coupled Fredholm type integral equations with the help of the Havelock’s inversion theorem and the porous plate condition. A multi-term Galerkin approximation in terms of Chebyshev polynomials is used to solve the vector integral equations and to obtain the numerical estimates for the reflection and the transmission coefficients. The computed numerical results for the reflection coefficient are depicted graphically for various values of several parameters. The present results are validated against the known results for the case of two identical impermeable plates and a single permeable plate submerged in water of finite depth.

Uniqueness of Solutions to First and Third Order Volterra Type Integral Equations on a Semiaxis

Sufficient conditions for the uniqueness of solutions to first and third kind Volterra integral equations in the space of functions continuous on the semiaxis are established in the case when the kernels of these equations can be alternating on the diagonal. Illustrative examples are given.

Coincidence and common fixed point theorems for four mappings satisfying (alpha(s),F)-contraction

In this paper, we manifest some coincidence and common fixed point theorems for four self-mappings satisfying Círíc-type and Hardy–Rogers-type (αs,F)-contractions defined on an αs-complete b-metric space. We apply these results to infer several new and old corresponding results in ordered b-metric spaces and graphic b-metric spaces. Our work generalizes several recent results existing in the literature. We present examples to validate our results. We discuss an application of main result to show the existence of common solution of the system of Volterra type integral equations.