Volterra Integral Equations 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.

Numerical Solution of Volterra's Integral Equations by Collocation and Taylor Method

In this article, two numerical methods (collocation method and Taylor method) are introduced to solve Volterra integral equation system. In the collocation method, using Bessel polynomials and collocation points, we convert the device into a matrix form and solve the device using the matrix form and obtain an approximate solution for the device. This answer is such that the larger the N becomes, the approximate answer is closer to the exact answer of the device. In Taylor method, With the use of the Taylor series, the system of integral equations is transformed into a matrix equation, and by integrating the output of this new system, a system of algebraic equations is created. By working through this device, we can obtain a rough solution for the Integral Equations device. Using these two methods when the answers are polynomial, we can get real answers.

Sharp Error Bounds for a Fractional Collocation Method for Weakly Singular Volterra Integral Equations with Variable Exponent

Sharp Error Bounds for a Fractional Collocation Method for Weakly Singular Volterra Integral Equations with Variable Exponent

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.

A class of monotonicity-preserving variable-step discretizations for Volterra integral equations

A class of monotonicity-preserving variable-step discretizations for Volterra integral equations

On the regularity of solutions to a class of nonlinear Volterra integral equations with singularities

We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval [0,b]. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on (0,b], with possible singularities of the derivatives of the solution at the left endpoint of the interval [0,b].

On solving 2D weakly singular Volterra integral equations of the second kind

On solving 2D weakly singular Volterra integral equations of the second kind

On Some Linear Two-Dimensional Volterra Integral Equations of the First Kind

On Some Linear Two-Dimensional Volterra Integral Equations of the First Kind

Stochastic Volterra equations for the local times of spectrally positive stable processes

Stochastic Volterra equations for the local times of spectrally positive stable processes

A multiparametric family of solutions to the Volterra linear integral equation of the first kind

We study the Volterra integral equation of the first kind with an integral operator of order n, a singularity and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms on the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term, we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows one to construct a particular and general solutions to the integro-differential equation in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multi-parameter family of solutions is being constructed for the original integral equation.

Boubaker operational matrix method for solving fractional weakly singular two-dimensional partial Volterra integral equation

Boubaker operational matrix method for solving fractional weakly singular two-dimensional partial Volterra integral equation

A Novel Super-Convergent Numerical Method for Solving Nonlinear Volterra Integral Equations Based on B-Splines

A Novel Super-Convergent Numerical Method for Solving Nonlinear Volterra Integral Equations Based on B-Splines

Radial solutions of a pseudo-differential equations associated with a p-adic non-local ultradiffusion operator

This paper is dedicated to study some pseudo-differential equations related to the operator Dd1,d2α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{\\alpha }_{d_1,d_2}$$\\end{document} with symbol Ad1,d2α(x)=max|x|pd1,|x|pd2α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}^{\\alpha }_{d_1,d_2}(x)=\\max \\left\\{ |x|_p^{d_1},\\ |x|_p^{d_2}\\right\\} ^\\alpha $$\\end{document}, acting on a radial function φ(|x|p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (|x|_p)$$\\end{document}. The importance of this symbol is that it is a generalization of Vladimirov and Bessel operators. Our research provides conditions for the existence of its right inverse and the explicit formulae for both operators, as well as a practical application. The inverse allows solving a Cauchy problem with the operator, which can be reduced to an integral equation whose properties are similar to those of the classical Volterra equation. We also find conditions for the existence and uniqueness of a mild solution for a degenerate nonlinear Cauchy problem.

Non-Linear Creep-Relaxation Constitutive Damage Model for Aging Concrete

A thermodynamic constitutive damage model for plain concrete, and other quasi-brittle aging materials, under creep relaxation is developed. The model accounts for the anisotropic damage induced through a second-order tensor damage variable. The aging viscoelasticity of the material is considered through the theory of solidification for aging solidifying materials. The material is considered a viscoelastic-damageable material. The Helmholtz free energy, utilized in the formulation, is treated based on the representation theorem of coupled damage strain tensors and Volterra integral equations. The model can analyze time-dependent damage (tertiary creep) under constant loading and can account for damage due to cyclic creep. Theoretical case studies are considered to illustrate the applicability of the model. The determination of the functions and constants, representing the material behavior, as well as any experimental companion is proposed for further research.

Regularization of Solutions of One Class of Systems of Linear Volterra Integral Equations of the First Kind in the Space of Differentiable Vector Functions on the Axis

Regularizing operators according to Lavrentiev are constructed and sufficient conditions for the uniqueness of solutions to systems of linear Volterra integral equations of the first kind on the axis are established.

Boundary Value Problems for a Model Hyperbolic Equation of the Third Order with Non-local Conditions of the First Kind

Considers nonlocal problems of the first kind for a model third-order hyperbolic equation. The main goal of the article is to prove the solvability of nonlocal problems of the first kind for a model third-order hyperbolic equation. Using the Riemann function method, the problem is reduced to Volterra integral equations of the second kind. Using the method of integral equations, the existence of a unique solution to nonlocal problems of the first kind is proven. The resulting solution to nonlocal problems of the first kind makes it possible to describe the process of moisture transfer in soils, heat transfer in a heterogeneous medium, and fluid filtration in porous media.

An innovative iterative approach to solving Volterra integral equations of second kind

Many scientists have shown great interest in exploring the realm of second-kind integral equations, offering many techniques for solving them, including exact, approximate, and numerical methods. This paper introduces the Hussein-Jassim method (HJ-method) for solving Volterra integral equations of the second kind (VIESKs). The foundation of this approach lies in the principle of Maclaurin expansion. The algorithm of the method was derived, and its convergence was analysed. Furthermore, the method was applied to various Volterra integral equations, encompassing linear, nonlinear, homogeneous, and nonhomogeneous cases. Ultimately, the proposed method successfully addressed these equations, with the approximate solutions converging toward the exact solution.

Weekly Ill-posed integral geometry problems of Volterra type in three-dimensional space

In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space.

Numerical methods for highly oscillatory Volterra integral equations with general oscillators

In this paper, collocation methods for linear Volterra integral equations with highly oscillatory trigonometric kernels are considered. Different from the cases with linear oscillator, we need a further action to obtain a practical scheme for general oscillators. To deal with it, the moment free Filon type method is employed in the exact collocation equation resulting to a fully discretized one. In order to conduct the convergence analysis, we firstly study the highly oscillatory integrals occurring in the error equations. Both the theoretical and numerical results show that our method not only converges with respect to step length but also has an asymptotic order which could be two. The asymptotic order indicates that the numerical results will become more accurate as the increase of frequency.

Collocation methods for nonlinear Volterra integral equations with oscillatory kernel

This work is devoted to the numerical solution of second kind nonlinear Volterra integral equations with highly oscillatory kernel. We use a collocation approach by discretizing the oscillatory integrals in the collocation equation using a Filon-type quadrature rule. We investigate the convergence of the numerical method in terms of step length h and frequency ω. As h decreases, the suggested technique converges with order d, while its asymptotic order as the frequency increase, is at least 1 and may reach 2 in some cases. Numerical experiments validate theoretical results.