Integral Equation Problems Research Articles

Linear–quadratic stochastic Volterra controls I: Causal feedback strategies

In this paper, we formulate and investigate the notion of causal feedback strategies arising in linear–quadratic control problems for stochastic Volterra integral equations (SVIEs) with singular and non-convolution-type coefficients. We show that there exists a unique solution, which we call the causal feedback solution, to the closed-loop system of a controlled SVIE associated with a causal feedback strategy. Furthermore, introducing two novel equations named a Type-II extended backward stochastic Volterra integral equation and a Lyapunov–Volterra equation, we prove a duality principle and a representation formula for a quadratic functional of controlled SVIEs in the framework of causal feedback strategies.

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

In this paper, the notion of singular backward stochastic Volterra integral equations (singular BSVIEs for short) in infinite dimensional space is introduced, and the corresponding well-posedness is carefully established. A class of singularity conditions are proposed, which not only cover that of fractional kernel, Volterra Heston model kernel, completely monotone kernels, to mention a few, but also happen to be used in the forward stochastic Volterra integral with new conclusions arising. Motivated by mathematical physics problem such as the viscoelasticity/thermoviscoelasticity of materials, heat conduction in materials with memory, optimal control problems of abstract stochastic Volterra integral equations (including fractional stochastic evolution equations and stochastic evolutionary integral equations) are presented. At last, our BSVIEs are surprisingly used in maximum principle of controlled stochastic delay evolution equations. One advantage of this new standpoint is that the final cost functional can naturally depend on the past state.

Linear-quadratic stochastic volterra controls II. Optimal strategies and riccati-volterra equations

In this paper, we study linear-quadratic control problems for stochastic Volterra integral equations with singular and non-convolution-type coefficients. The weighting matrices in the cost functional are not assumed to be non-negative definite. From a new viewpoint, we formulate a framework of causal feedback strategies. The existence and the uniqueness of a causal feedback optimal strategy are characterized by means of the corresponding Riccati-Volterra equation. The causal feedback optimal strategy is explicitly written by a finite dimensional (matrix-valued) function which solves the Riccati-Volterra equation.

A meshfree approach for vibro-acoustic analysis of combined composite laminated shells with variable thickness

The vibro-acoustic responses of combined composite laminated shell structures with variable thickness immersed in acoustic medium is investigated by using a meshfree method in this paper. The theoretical model of structural field is established by employing the energy principle in conjunction with the first-order shear deformation theory (FSDT) and the theoretical model of acoustic field are established by using Kirchhoff-Helmholtz integral equation. The displacement components of shell segments and the sound pressure variables of acoustic medium are described by using the meshfree Tchebychev point interpolation (TPIM) shape function and Double Fourier series in meridional and circumferential directions. The non-uniqueness problem of Kirchhoff-Helmholtz integral equation is overcome by adopting CHIEF method. The validation of the presented theoretical model is proved by comparing the results calculated by the proposed model with those results of existing literature and FEM/BEM. Based on the established theoretical model, vibro-acoustic responses of combined composite laminated shell structures with variable thickness immersed in acoustic medium are performed by analyzing the influences of thickness parameter, external load and boundary condition on the sound pressure level (SPL) and sound power level (SWL) of combined composite laminated shell structures. The above investigations can provide the theoretical thesis and technique guidance of preliminary design of combined composite laminated shell immersed in various acoustic medium.

The approximate solution of nonlinear heat equation

AbstractNowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type.

On the Maximum Principle for Optimal Control Problems of Stochastic Volterra Integral Equations with Delay

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.

Gaussian entanglement witness and refined Werner-Wolf criterion for continuous variables

We use matched quantum entanglement witnesses to study the separable criteria of continuous variable states. The witness can be written as an identity operator minus a Gaussian operator. The optimization of the witness then is transformed to an eigenvalue problem of a Gaussian kernel integral equation. It follows a separable criterion not only for symmetric Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from symmetric Gaussian states. Based on Fock space numeric calculation, we obtain an entanglement witness for more general two-mode states. A necessary criterion of separability follows for two-mode states and it is shown to be necessary and sufficient for a two mode squeezed thermal state and the related two-mode non-Gaussian states. We also connect the witness based criterion with Werner-Wolf criterion and refine the Werner-Wolf criterion.

Coincidence Theorems under Generalized Nonlinear Relational Contractions

After the appearance of relation-theoretic contraction principle due to Alam and Imdad, the domain of fixed point theory applied to relational metric spaces has attracted much attention. Existence and uniqueness of fixed/coincidence points satisfying the different types of contractivity conditions in the framework of relational metric space have been studied in recent times. Such results have the great advantage to solve certain types of matrix equations and boundary value problems for ordinary differential equations, integral equations and fractional differential equations. This article is devoted to proving the coincidence and common fixed point theorems for a pair of mappings (T,S) employing relation-theoretic (ϕ,ψ)-contractions in a metric space equipped with a locally finitely T-transitive relation. Our results improve, modify, enrich and unify several existing coincidence points as well as fixed point results. Several examples are provided to substantiate the utility of our results.

Analysis of the family of integral equation involving incomplete types of I and Ī-functions

ABSTRACT The present article introduces and studies the Fredholm-type integral equation with an incomplete I-function (IIF) and an incomplete -function (I F) in its kernel. First, using fractional calculus and the Mellin transform principle, we solve an integral problem involving IIF. The idea of the Mellin transform and fractional calculus is then used to analyse an integral equation using the incomplete -function. This is followed by the discovery and investigation of several important exceptional cases. This article's general discoveries may yield new integral equations and solutions. The desired outcomes seem to be very helpful in resolving many real-world problems whose solutions represent different physical phenomena. And also, findings help solve introdifferential, fractional differential, and extended integral equation problems.

Solution of Linear Volterra Integral Equation of Second Kind via Rishi Transform

The solution of various problems of engineering and science can easily determined by representing these problems in integral equations. There are numerous analytical and numerical methods which can be used for solving different kinds of integral equations. In this paper, authors used recently developed integral transform “Rishi Transform” for obtaining the analytical solution of linear Volterra integral equation of second kind (LVIESK). For this, the kernel of LVIESK has assumed a convolution type kernel. Five numerical examples are considered for demonstrating the complete procedure of determining the solution. Results of these problems suggest that Rishi transform provides the exact analytical solution of LVIESK without doing complicated calculation work.

Causal state feedback representation for linear quadratic optimal control problems of singular Volterra integral equations

<p style='text-indent:20px;'>This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Our framework covers the problems for fractional differential equations. Under some necessary convexity conditions, an optimal control exists, and can be characterized via Fréchet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations are not causal, meaning that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. We obtain a causal state feedback representation of the optimal control via a Fredholm integral equation. Finally, a concrete form of our results for fractional differential equations is presented.</p>

Integral equation analysis of multiport H‐plane microwave circuits by using 2D rectangular cavity Green's functions accelerated by the Ewald Method

An integral equation technique for the analysis of multiport H-plane microwave circuits composed of an unlimited number of arbitrarily shaped metallic and/or dielectric elements is proposed. The structure under study is divided into different regions after the application of the surface equivalence principle. Following this approach, the access waveguide ports are modelled by means of parallel plate Green's functions, whereas the central region is characterised by 2D rectangular cavity Green's functions. The use of this kind of Green's functions represents a novelty for solving H-plane integral equation problems. For the first time, the Ewald method has been employed in order to accelerate the computation of the 2D rectangular cavity Green's functions and their derivatives, needed in the integral equation. In addition, convergence studies show the numerical convenience of using the Ewald method to achieve a fast and accurate evaluation of these Green's functions. Results show that the analysis can be very efficient if the structure is segmented in a proper way based on fitting the cavity resonator within the geometry of the structure under analysis. In fact, all the walls of the structure that are coincident with the auxiliary resonator need not to be discretised during the numerical solution. Finally, several simulation examples of practical inductive microwave circuits are presented and discussed, showing a good agreement and higher computational efficiency when compared to results provided by commercial full-wave software tools and measurements.

Overdetermined problems for negative power integral equations on bounded domain

The aim of this paper is to study symmetry and monotonicity of positive solutions for the following overdetermined problem where , p>0, A>0, , is a bounded domain. We first prove that on if and only if Ω is a ball. Next we consider the partially overdetermined problem. If Γ is a proper open set of and u = C in , we show that under some assumptions on the geometry of Γ, Ω is a ball. Furthermore, we derive that all positive solutions of above equations are radially symmetric and monotone increasing with respect to the radius by using the method of moving planes.

On the spurious resonance modes of time domain integral equations for analyzing acoustic scattering from penetrable objects.

The interior resonance problem of time domain integral equations (TDIEs) formulated to analyze acoustic field interactions on penetrable objects is investigated. Two types of TDIEs are considered: The first equation, which is termed the time domain potential integral equation (TDPIE), suffers from the interior resonance problem, i.e., its solution is replete with spurious modes that are excited at the resonance frequencies of the acoustic cavity in the shape of the scatterer. Numerical experiments demonstrate that, unlike the frequency-domain integral equations, the amplitude of these modes in the time domain could be suppressed to a level that does not significantly affect the solution. This is achieved by increasing the numerical solution accuracy through the use of a higher-order discretization in space and the band limited approximate prolate spheroidal wave function with high interpolation accuracy as basis function in time. The second equation is obtained by linearly combining TDPIE with its normal derivative. The solution of this equation, which is termed the time domain combined potential integral equation (TDCPIE), does not involve any spurious interior resonance modes but it is not as accurate as the TDPIE solution at non-resonance frequencies. In addition, TDCPIE's discretization calls for treatment of hypersingular integrals.

On Global Convergence of Third-Order Chebyshev-Type Method under General Continuity Conditions

There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method is proposed under the assumptions that Fréchet derivative of first order satisfies continuity condition of the Hölder. Finally, we consider some integral equation and boundary value problem (BVP) in order to illustrate the suitability of theoretical results.

Mixed Problems for Plane Parabolic Systems and Boundary Integral Equations

We consider the mixed problem for a one-dimensional (with respect to the spatial variable) second-order parabolic system with Dini-continuous coefficients in a domain with nonsmooth lateral boundaries. Using the method of boundary integral equations, we find a classical solution of this problem. We investigate the smoothness of the solution as well.

Existence, uniqueness, Ulam–Hyers–Rassias stability, well-posedness and data dependence property related to a fixed point problem in gamma-complete metric spaces with application to integral equations

In this paper, we study a fixed point problem for certain rational contractions on γ-complete metric spaces. Uniqueness of the fixed point is obtained under additional conditions. The Ulam–Hyers–Rassias stability of the problem is investigated. Well-posedness of the problem and the data dependence property are also explored. There are several corollaries of the main result. Finally, our fixed point theorem is applied to solve a problem of integral equation. There is no continuity assumption on the mapping.

Stability and convergence analysis of hybrid algorithms for Berinde contraction mappings and its applications

In this paper, we construct a new hybrid iteration, called SR-iteration, and prove its stability and convergence analysis for weak contraction mappings in a Banach space. We compare rate of convergence between the SR-iteration and other iterations. Moreover, we provide numerical comparisons for supporting our main theorem and apply our main result to prove existence problem of mixed type Volterra-Fredholm functional nonlinear integral equation.

A Pretreatment Method of Volterra the External Boundary Value Problem of Integral Differential Equations

In the process of traditional methods, the error rate of external boundary value problem is always at a high level, which seriously affects the subsequent calculation and cannot meet the requirements of current Volterra products. To solve this problem, Volterra's preprocessing method for the external boundary value problem of Integro differential equations is studied in this paper. The Sinc function is used to deal with the external value problem of Volterra Integro differential equation, which reduces the error of the external value problem and reduces the error of the external value problem. In order to prove the existence of the solution of the differential equation, when the existence of the solution can be proved, the differential equation is transformed into a Volterra integral equation, the Taylor expansion equation is used, the symplectic function is used to deal with the external value problem of homogeneous boundary conditions, and the uniform effective numerical solution of the external value problem of the equation is obtained by homogeneous transformation according to the non-homogeneous boundary conditions.

Solvability of some nonlinear boundary value problems for singular integral equations of convolution type

Solvability of some nonlinear boundary value problems for singular integral equations of convolution type