Dimensional Volterra Research Articles

Numerical solution and dynamical studies for (2 + 1) dimensional Volterra–Fredholm integral equations with a discontinuous kernel

Numerical solution and dynamical studies for (2 + 1) dimensional Volterra–Fredholm integral equations with a discontinuous kernel

Lyapunov functions and dynamics of infinite dimensional Volterra operators

The majority of research on quadratic stochastic operators (QSOs) was done on a finite-dimensional set of all probability distributions (also known as a simplex), thus it is intriguing to extend them for an infinite case. In particular, the infinite dimensional Volterra operators are discussed in the current paper. Due to the fact non-compactness of infinite dimensional simplex (unlike a finite case) makes the general study challenging, therefore, a sub-class of infinite dimensional Volterra operators is introduced. Furthermore, we construct Lyapunov functions which allow us to explore the dynamics of the introduced operators. Several examples are given with a full description of the limiting set.

A 2+1 dimensional Volterra type system with nonzero boundary conditions via Dbar dressing method

A 2+1 dimensional Volterra type system with nonzero boundary conditions via Dbar dressing method

Two dimensional Volterra integral equation with singular kernels in contact problems and its numerical computations

Two dimensional Volterra integral equation with singular kernels in contact problems and its numerical computations

Infinite dimensional pathwise Volterra processes driven by Gaussian noise – Probabilistic properties and applications –

We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Holder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from [18] to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al. in [13].

On omega limiting sets of infinite dimensional Volterra operators

In the present paper, we are aiming to study limiting behaviour of infinite dimensional Volterra operators. We introduce two classes and of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets ωV and with respect to ℓ1-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to , then the sets and coincide for every x ∈ S, and moreover, they are non empty. If Volterra operator belongs to , then ωV(x) could be empty, and it implies the non-ergodicity (w.r.t. ℓ1-norm) of V, while it is weak ergodic.

Solving Three Dimensions Volterra Integral Equations (TDVIE) via a Neural Network

The aim of this paper is present a new numerical method for solvingThree Dimensions Volterra Integral Equations using artificial neural network by design multilayer feed forward Neural Network. A multi- layers design in our proposed method consist of a hidden layer having seven hidden units. and one linear output unit. Linear Transfer function used as each unit and using Levenberg- Marquardtalgorithmtraining. Moreover, examples on three- dimensional Volterra integral equations carried out to illustrate the accuracy and the efficiency of the presented method. In addition, some comparisons among proposed method and Shifted Chebyshev Polynomials method and Reduced Differential Transform Method are presented

Comparison of Chebyshev and Legendre polynomials methods for solving two dimensional Volterra - Fredholm integral equations

Abstract In this paper we apply Chebyshev and Legendre polynomials methods to obtain the approximate solution of mixed Volterra - Fredholm singular integral equations. Comparison between the two methods has been presented, we present some examples to determine the accuracy of the methods.

Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system

In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N. We derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmannians GrR(k,N) and GrC(k,N).

A computational method for solving a class of two dimensional Volterra integral equations

The use of the Geometric Series theorem, together with Schauder bases in certain Banach spaces, allows us to design a numerical algorithm in order to solve an important type of two-dimensional linear integral equations. A study of the convergence and error is presented here. All the calculations can be easily implemented and the efficiency of this method will be shown with numerical results.

Solving three-dimensional Volterra integral equation by the reduced differential transform method

In this article, the results of two-dimensional reduced differential transform method is extended to three-dimensional case for solving three dimensional Volterra integral equation. Using the described method, the exact solution can be obtained after a few number of iterations. Moreover, examples on both linear and nonlinear Volterra integral equation are carried out to illustrate the efficiency and the accuracy of the presented method.

Runge-Kutta and Block by Block Methods to Solve Linear Two-Dimensional Volterra Integral Equation with Continuous Kernel

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

Runge-Kutta and Block by Block Methods to Solve Linear Two-Dimensional Volterra Integral Equation with Continuous Kernel

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

AN APPROXIMATE SOLUTION OF TWO DIMENSIONAL NONLINEAR VOLTERRA INTEGRAL EQUATION USING NEWTON-KANTOROVICH METHOD

This paper studies the method for establishing an approximate solution of nonlinear two dimensional Volterra integral equations (NLTD-VIE). The Newton-Kantorovich (NK) suppositions are employed to modify NLTD-VIE to the sequence of linear two dimensional Volterra integral equation (LTDVIE). The proper-ties of the two dimensional Gauss-Legenre (GL) quadrature fromula are used to abridge the sequence of LTD-VIE to the solution of the linear algebraic system. The existence and uniqueness of the approximate solution is demonstrated, and an illustrative example is provided to show the precision and authenticity of the method. Keywords: Newton-Kantorovich method, nonlinear operator, two dimensional Volterra integral equation, two dimensional Gauss-Legendre formula.

A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations

For stably dissipative Lotka{Volterra equations the dynamics on the attractor are Hamiltonian and we argue that complex dynamics can occur. We also present examples and properties of some infinite dimensional Volterra systems with applications related with stably dissipative Lotka-Volterra equations. We finish by mentioning recent contributions on the subject.

One variant of a (2 + 1)-dimensional Volterra system and its (1 + 1)-dimensional reduction

A new system is generated from a multi-linear form of a (2+1)-dimensional Volterra system. Though the system is only partially integrable and needs additional conditions to possess two-soliton solutions, its (1+1)-dimensional reduction gives an integrable equation which has been studied via reduction skills. Here, we give this (1+1)-dimensional reduction a simple bilinear form, from which a Backlund transformation is derived and the corresponding nonlinear superposition formula is built.

Stability of perturbed [formula omitted]-dimensional Volterra differential equations

In this work we deal with the problem of the stability and uniform stability of the perturbed n -dimensional Volterra integral and differential equation x ′ ( t ) = A ( t ) x ( t ) + ∫ − ∞ t C ( t , s ) x ( s ) d s + ∫ − ∞ t D ( t , s ) x ′ ( s ) d s + b ( t ) . Some sufficient conditions for the zero solution of this equation to be stable as well as uniformly stable have been obtained.

A generalised integral quadratic method: improvement of the solution for one dimensional Volterra integral equation using particle swarm optimisation

In a previous work, we have introduced the generalised integral quadratic method (GIQ) to handle the Volterra's integral equations in which the interpolating points of Tchebychev have been used. In this paper the particle swarm optimisation (PSO) is considered to improve, qualitatively, the solutions. This new approach is investigated and tested with some numerical examples in the linear and nonlinear cases.

On infinite dimensional quadratic Volterra operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.

On some two dimensional Volterra type linear integral equations with super-singularity

On some two dimensional Volterra type linear integral equations with super-singularity