Mixed Volterra-Fredholm Integrodifferential Equations Research Articles

Approximate Solution of System of Multi-term Fractional Mixed Volterra–Fredholm Integro-Differential Equations by BPFs

Approximate Solution of System of Multi-term Fractional Mixed Volterra–Fredholm Integro-Differential Equations by BPFs

Existence and controllability of fractional semilinear mixed Volterra-Fredholm integro differential equation

In this paper, the existence of mild solution and controllability of fractional semilinear mixed Volterra–Fredholm integro differential equations of order 1<γ<2 with nonlocal conditions are investigated. First we discuss the existence of mild solution and then controllability of the proposed system by using the concept of Banach fixed point theorem and semigroup theory. For the authenticity and applicability of the presented theory some examples are also provided.

A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order $ {r} \in (1, 2) $ with impulses

&lt;abstract&gt;&lt;p&gt;In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order $ 1 &amp;lt; r &amp;lt; 2 $. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.&lt;/p&gt;&lt;/abstract&gt;

DISCRETE WEIGHTED RESIDUAL METHODS WHICH ARE TECHNIQUES USED FOR NUMERICAL SOLUTION TO MIXED VOLTERRA-FREDHOLM FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

This study aims to solve the most general form of linear mixed Volterra-Fredholm integro-differential equations of multi-fractional order in the Caputo sense (MV-FIFDEs), which is solved by using orthogonal generalized Bernstein’s polynomial expansion with collocation and moment in the discrete weighted residual method under suitable conditions; then, the Clenshaw-Curtis formula is applied to approximate the integral terms of the equation numerically. In this work, the integro-fractional differential equations are reduced to algebraic linear equations and then to an operational matrix, and the solution of the resultant system yields the unknown Bernstein coefficients of approximation solutions. An algorithm has been created for each technique to handle the MV-FIFDEs using the described methods. Furthermore, numerical examples are presented to demonstrate and compare the techniques’ validity and applicability and comparisons with previous results. The majority of programs are performed on a computer using MATLAB v. 9.7. Keywords: Mixed Fractional Integro-Differential Equations, Caputo Derivative, Bernstein Polynomial, Collocation, Moment, Discrete Weighted Residual, Clenshaw-Curtis Formula, Volterra-Fredholm Equations DOI： https://doi.org/10.35741/issn.0258-2724.58.3.43

Spectral Collocation Algorithm for Solving Fractional Volterra-Fredholm Integro-Differential Equations via Generalized Fibonacci Polynomials

In this research article, we build and implement an efficient spectral algorithm for handling linear/nonlinear mixed Volterra-Fredholm integro-differential equations. First, we expand the exact solution as a truncated series of the generalized Fibonacci polynomials, and then we discretize the equation via Simpson's quadrature formula. Finally, we collocate the resulted residual at the roots of the shifted first-kind Chebyshev polynomials. Also, the rate of convergence is studied and the truncation error estimate is reported. Some numerical examples are exhibited to prove the applicability and accuracy of the algorithm.

New results on nonlocal fractional Volterra-Fredholm integro-differential equations

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence and uniqueness of solutions. Results of this paper are based on nonstandard assumptions and hypotheses and provide a supplementary result concerning the regularity of solutions.

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

A Hybrid Approach of Nonlinear Partial Mixed Integro-Differential Equations of Fractional Order

In this paper, we use hybrid parabolic and block-pulse functions (2D-PBPFs) to provide an approximate solution of nonlinear partial mixed Volterra–Fredholm integro-differential equations of fractional order. To reach this goal, we present the Volterra integral operational matrix, operational matrix of fractional integral and operational matrix of mixed Volterra–Fredholm integral by 2D-PBPFs. Using the proposed method, nonlinear partial mixed Volterra–Fredholm integro-differential equations of fractional order become into a nonlinear system of algebraic equations. Moreover, we provide some theorems for convergence analysis and we demonstrate that the convergence order of the suggested approximate approach is $$O(h^{3})$$. Finally, we solve two numerical examples to prove the accuracy of the proposed method.

Numerical solution of the mixed Volterra–Fredholm integro-differential multi-term equations of fractional order

This paper aims to introduce two effective numerical approaches for solving the mixed Volterra–Fredholm integro-differential equation of fractional order and multi-order with initial values. Despite the fact that the second approach transfers the integro-differential equations into a system of algebraic equations through the usage of operational matrices, these problems can be transferred to a system of algebraic equations by expanding the solution’s highest order derivative through the block pulse functions (BPFs). This is done by using the generalized operational matrices of BPFs for differentiation along with the fractional calculus properties in the first scheme, in which convergence of the solution obtained has been shown in the following.The accuracy and applicability of two proposed approaches will be compared by some relevant numerical examples, for which the exact solution is known.

Numerical solution of nonlinear mixed Volterra-Fredholm integro-differential equations by two-dimensional block-pulse functions

AbstractThis paper proposed an effective numerical method to obtain the solution of nonlinear two-dimensional mixed Volterra-Fredholm integro-differential equations. For this purpose, the two-dimen...

A new sequential approach for solving the integro-differential equation via Haar wavelet bases

In this work, we present a method for numerical approximation of fixed point operator, particularly for the mixed Volterra–Fredholm integro-differential equations. The main tool for error analysis is the Banach fixed point theorem. The advantage of this method is that it does not use numerical integration, we use the properties of rationalized Haar wavelets for approximate of integral. The cost of our algorithm increases accuracy and reduces the calculation, considerably. Some examples are provided toillustrate its high accuracy and numerical results are compared with other methods in the other papers.

An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two-Dimensional Volterra-Fredholm Integrodifferential Equations

We propose and apply coupling of the variational iteration method (VIM) and homotopy perturbation method (HPM) to solve nonlinear mixed Volterra-Fredholm integrodifferential equations (VFIDE). In this approach, we use a new formula called variational homotopy perturbation method (VHPM) and variational accelerated homotopy perturbation method (VAHPM). This approach is based on the form of He’s polynomials and on a new form of He’s polynomials. We discuss the convergence of the technique. Some numerical examples are introduced to verify the efficiency of this technique.

Nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition

The aim of the present paper is to investigate the global existence of mild solutions of nonlinear mixed Volterra-Fredholm integrodifferential equations, with nonlocal condition. Our analysis is based on an application of the Leray-Schauder alternative and rely on a priori bounds of solutions.

Existence of Solutions to Fractional Mixed Integrodifferential Equations with Nonlocal Initial Condition

We study the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition , where , , and is a given function. We point out that such a kind of initial conditions or nonlocal restrictions could play an interesting role in the applications of the mentioned model. The results obtainded are applied to an example.

Nonlocal Cauchy problem for nonlinear mixed integrodifferential equations

The present paper investigates the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition. The results obtained by using Schauder fixed point theorem and the theory of semigroups.

Controllability of mixed Volterra–Fredholm-type integro-differential systems in Banach space

Dhakne and Kendre [On abstract nonlinear mixed Volterra–Fredholm integro-differential equations, Presented Paper in the International Conference at IIT-Bombay, 11–13 December, 2004] has proved the existence of the abstract nonlinear mixed Volterra–Fredholm integro-differential system of the type x ′ ( t ) = f t , x ( t ) , ∫ 0 t k ( t , s , x ( s ) ) d s , ∫ 0 T h ( t , s , x ( s ) ) d s , x ( 0 ) = x 0 ∈ X ; t ∈ J = [ 0 , T ] . In this short article, we have studied sufficient conditions for controllability of semi-linear mixed Volterra–Fredholm-type integro-differential systems in Banach space of the type x ′ ( t ) = Ax ( t ) + ( Bu ) ( t ) + f t , x ( t ) , ∫ 0 t g ( t , s , x ( s ) ) d s , ∫ 0 T h ( t , s , x ( s ) ) d s , x ( 0 ) = x 0 , t ∈ J = [ 0 , T ] , where the state x ( . ) takes values in a Banach space X and the control function u ( . ) is given in L 2 ( J , U ) , with U as a Banach space. Here A is the infinitesimal generator of a strongly continuous semigroup in a Banach space X . B is a bounded linear operator from U into X . The result is obtained by using the application of the topological transversality theorem known as Leray–Schauder alternative and rely on a priori bounds of solutions. An example is provided to illustrate the theory.