Volterra-Fredholm Type Research Articles

Solving Linear Volterra-Fredholm Integro-Differential Equations using Chebyshev-Galerkin with Error Estimation

The semi-analytic method based on the Galerkin technique with Chebyshev basis is presented in this paper to solve the linear integro-differential equations in the Volterra-Fredholm type. To facilitate, the use of the Chebyshev-Galerkin technique, new theorems and lemmas are established. Different numerical examples are introduced to demonstrate how the proposed method has efficacy and is easier to apply for this problem type. Some issues’ domains don’t match the Chebyshev’s domain. So, how to adequate the solution domain is presented. The error estimate is computed to prove the method's applicability and accuracy of the proposed method for any problem, especially in the case of the exact solution is not easy to determine.

Approximate controllability of third order dispersion systems

The current paper focuses on the approximate controllability of Volterra-Fredholm type integrodifferential third order dispersion system. The Banach fixed point theorem and semigroup operator are used to obtain the existence and uniqueness of mild solutions to the considered system. In order to examine the approximate controllability results, a different approach is used, namely, the sequential method.

Existence of solution for Volterra–Fredholm type stochastic fractional integro‐differential system of order μ ∈ (1, 2) with sectorial operators

The mainspring of the study is to investigate the out‐turn of stochastic Volterra–Fredholm integro‐differential inclusion of order with sectorial operator of the type . The existence results of our proposed problem is derived by employing Martelli's fixed point approach. We do not limit the theoretical results of fractional stochastic equation to local condition but extend to nonlocal condition, and physical interpretation of our obtained results is proved with an appropriate illustration.

Approximate solutions of a certain initial value problem

This work aims to discuss the approximation of an initial value problem for integro-differential equation (IDEqs). of Volterra-Fredholm type (V-FIDEq), with given initial conditions (IC). These solutions are a class of conditions functions C(R+ Rn), where R+, = [0, ∞), and Rn represents the n-dimensional Euclidean space and stands in for the norm |.|. We are presenting several types of particular inequality with candid appreciation. These types are applied to find out the results of this paper.

TWO NUMERICAL METHODS FOR SOLVING A NONLINEAR SYSTEM OF INTEGRAL EQUATIONS OF MIXED VOLTERRA-FREDHOLM TYPE ARISING FROM A CONTROL PROBLEM RELATED TO LEUKEMIA

The aim of this paper is to present two algorithms for numerical solving of a fixed final state control problem in connection with the leukemia treatment strategy. In the absence of the controllability condition, our model leads to a nonlinear integral system of Volterra type to whom explicit iterative techniques apply and converge. Once using the controllability condition, the control variable is expressed in terms of the state variables and the integral system changes to a mixed Volterra-Fredholm type one making direct iterative techniques inoperative. However, two paths can be followed. One consists in an iterative procedure where at each step the control variable is calculated using the approximate values of the state variables from the previous step. The other looks for the numerical value of the control variable by using the bisection method. Numerical simulations, error analysis and biological interpretation are given.

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

Optimal control results for Sobolev‐type fractional mixed Volterra–Fredholm type integrodifferential equations of order 1 < r < 2 with sectorial operators

AbstractThe goal of this research is to investigate the issue of existence and optimal control results for fractional mixed Volterra–Fredholm type integrodifferential systems of order with sectorial operators. Initially, we use fractional derivatives, semigroup theory, mild solutions, Sobolev‐type, and Schauder's fixed point theorem to examine the existence of results. Then, we will move on to the systems with optimal control results. Finally, we show a theoretical application to establish the validity of the findings.

Fixed point Theorem for Interpolative Mappings in F-Mv -Metric Space with an Application

Abstract The aim of this paper is to prove fixed point results for Interpolativemappings in F-Mv -metric spaces with an application which cannot be obtained from the corresponding results in metric spaces. We also provide an illustrative example to support our results. Besides discussing an application to Volterra-Fredholm type integral equations.

Legendre-Galerkin method for the nonlinear Volterra-Fredholm integro-differential equations

The study of solving nonlinear integro-differential equations in Volterra-Fredholm type presents in this paper. The proposed method tends to use Legendre polynomials as a basis in the Galekin method to obtain the numerical solution. We use the Newton method to get the numerical solution of the nonlinear equations resulted from applying the Galerkin method. The comparison of the present study with the existing results in the literature shows an excellent agreement. Numerical examples explain the convergence, applicability, and efficiency of algorithm.

On some new nonlinear Volterra-Fredholm type discrete inequalities and its applications

On some new nonlinear Volterra-Fredholm type discrete inequalities and its applications

HALF-LINEAR VOLTERRA-FREDHOLM TYPE INTEGRAL INEQUALITIES ON TIME SCALES AND THEIR APPLICATIONS

The main aim of this paper is to establish some new half-linear Volterra-Fredholm type integral inequalities on time scales. Our results not only extend and complement some known integral inequalities but also provide an effective tool for the study of qualitative properties of solutions of some dynamic equations.

Fixed Points of Subadditive Maps with an Application to a System of Volterra–Fredholm Type Integrodifferential Equations

In this paper, some fixed-point theorems are established for strongly subadditive maps on CΩ,ϒ (where CΩ,ϒ denotes the space of ϒ-valued continuous functions on a compact Hausdorff space Ω and ϒ is a unital Banach algebra). Finally, the result is applied to prove the existence and uniqueness of a solution for a system of nonlinear integrodifferential equations.

Some Nonlinear Delay Volterra–Fredholm Type Dynamic Integral Inequalities on Time Scales

We are devoted to studying a class of nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales, which can provide explicit bounds on unknown functions. The obtained results can be utilized to investigate the qualitative theory of nonlinear delay Volterra–Fredholm type dynamic equations. An example is also presented to illustrate the theoretical results.

A class of retarded Volterra-Fredholm type integral inequalities on time scales and their applications

In this paper, we study some new retarded Volterra-Fredholm type integral inequalities on time scales, which provide explicit bounds on unknown functions. These inequalities generalize and extend some known inequalities and can be used as tools in the qualitative theory of certain classes of retarded dynamic equations on time scales. Some applications are also presented to illustrate the usefulness of our results.

Estimates on some power nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales

In this paper, some explicit bounds on solutions to a class of new power nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales are established, which can be used as effective tools in the study of certain dynamic equations. Application examples are also given.

Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications

In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations.

Some new Volterra-Fredholm type discrete inequalities with four iterated infinite sums with applications

Some new Volterra-Fredholm type discrete inequalities with four iterated infinite sums with applications

Some new generalized Volterra-Fredholm type discrete fractional sum inequalities and their applications

In this paper, we present some new Volterra-Fredholm-type discrete fractional sum inequalities. These inequalities can be used as handy and powerful tools in the study of certain fractional sum-difference equations. Some applications are also presented to illustrate the usefulness of our results.

Some Volterra-Fredholm type nonlinear inequalities involving four iterated infinite integral and application

Some Volterra-Fredholm type nonlinear inequalities involving four iterated infinite integral and application

Estimation on solutions of a class of generalized nonlinear Volterra-Fredholm type integral inequalities with applications

Estimation on solutions of a class of generalized nonlinear Volterra-Fredholm type integral inequalities with applications