Fuzzy Volterra Integral Equations Research Articles

SOLUTION OF FUZZY VOLTERRA INTEGRAL EQUATIONS USING COMPOSITE METHOD

With the rapid development of of fuzzy Integral equations, the problem of finding its solution becomes increasingly important for knowledge propagation and putting researchers wisdom to work. A recent development trend of fuzzy integral equations is modeling many problems in the field of science and technology. If the unknown function in the considered equation has a solution in terms of infinite series expansion, this proposed method becomes more accurate to find the exact solution. According on the parametric form of a fuzzy number, a fuzzy integral equation converts to two systems of integral equations in the crisp case, and then proposed method is achieved to obtain the exact fuzzy solutions of fuzzy Volterra integral equations. To validate the effectiveness of proposed method, some examples of considered equations are solved and the results show that it achieved the best performance.

Approximate Analytical Solution of Fuzzy Linear Volterra Integral Equation via Elzaki ADM

In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy solutions are determined using Elzaki ADM after the fuzzy linear Volterra integral equations are first translated into two crisp integral equations utilizing the fuzzy number in parametric form. Three instances of the considered equations are solved to show the established scheme’s dependability, efficacy, and application. The results have a substantial impact on the fuzzy analytical dynamic equation theory. The comparison of the data in a graphical and tabular format demonstrates the robustness of the defined regime. The lower and upper bound solutions’ theoretical convergence and error estimates are highlighted in this paper. A tolerable order of absolute error is also obtained for this inquiry, and the consistency of the outcomes that are approximated and accurate is examined. The regime generated effective and reliable results. The current regime effectively lowers the computational cost, and a faster convergence of the series solution to the exact answer is signaled.

A study on fuzzy Volterra integral equations for [formula omitted]-correlated fuzzy processes on time scales

In the paper, we propose a new notion of differentiability and integrability for S-correlated fuzzy processes on time scales and highlight some of their properties. With these concepts, we investigate fuzzy Volterra integral equations for S-correlated fuzzy processes on time scales. Also, we present a connection between fuzzy Volterra integral equations and fuzzy initial value problems for S-correlated fuzzy processes on time scales. In particular, we establish a fuzzy Adomian decomposition method to solve fuzzy Volterra integral equations with fuzzy kernel on time scales. Some illustrative examples are given.

Approximation Solution of Fuzzy Singular Volterra Integral Equation by Non-Polynomial Spline

A non-polynomial spline (NPS) is an approximation method that relies on the triangular and polynomial parts, so the method has infinite derivatives of the triangular part of the NPS to compensate for the loss of smoothness inherited by the polynomial. In this paper, we propose polynomial-free linear and quadratic spline types to solve fuzzy Volterra integral equations (FVIE) of the 2nd kind with the weakly singular kernel (FVIEWSK) and Abel's type kernel. The linear type algorithm gives four parameters to form a linear spline. In comparison, the quadratic type algorithm gives five parameters to create a quadratic spline, which is more of a credit for the exact solution. These algorithms process kernel singularities with a simple technique. Illustrative examples use MathCad software to deal with upper and lower-bound solutions to fuzzy problems. The method provides a reliable way to ensure that an exact solution is approximated. Also, figures and tables show the potential of the method.

On the Solution of Fuzzy Volterra Integral Equation with Piecewise Kernel

On the Solution of Fuzzy Volterra Integral Equation with Piecewise Kernel

Solution of Fuzzy Volterra Integral and Fractional Differential Equations via Fixed Point Theorems

In this paper we present fuzzy coupled fixed point results in the turf of complete b-metric spaces via nonlinear F-contraction; in follow we derive some interesting results as byproducts. Eventually, we apply our results in solving fuzzy Volterra integral equations and Caputo-Hadamard type of fractional differential equations.

Numerical solution of a linear fuzzy Volterra integral equation of the second kind with weakly singular kernels

Numerical solution of a linear fuzzy Volterra integral equation of the second kind with weakly singular kernels

New Techniques for Solutions of Fuzzy Volterra Integral Equations via Soft Set-Valued Maps

In this paper, new generalized concepts of set-valued maps under the name soft set-valued maps are established. The presented results herein are soft set extensions of various existing fuzzy set-valued and crisp multivalued fixed point theorems. Examples are provided to authenticate the hypotheses of our obtained results. Moreover, our main result is applied to establish conditions for existence of solutions to fuzzy Volterra integral equations.

Usage of the Fuzzy Laplace Transform Method for Solving One-Dimensional Fuzzy Integral Equations

In this paper, we propose the solution of fuzzy Volterra and Fredholm integral equations with convolution type kernel using fuzzy Laplace transform method (FLTM) under Hukuhara differentiability. It is shown that FLTM is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated with few examples to show the ability of the proposed method.

Iterative numerical method for pantograph type fuzzy Volterra integral equations

In this paper a sequence of fuzzy Bernstein spline interpolation procedure is considered for approximating the solution of pantograph type fuzzy Volterra integral equation. The numerical method is based on an iterative algorithm that applies at each iterative step the fuzzy Bernstein spline interpolation procedure and the Bernstein composite quadrature formula. The error estimate given in terms of the Lipschitz constants is used for obtaining the convergence of the iterative method. The theoretical results are tested on some numerical examples illustrating the accuracy of the method.

Iterative approach for a class of fuzzy Volterra integral equations using block pulse functions

Fuzzy Integral equations is a mathematical tool for mod- eling the uncertain control system and economic. In this paper, we present numerical solution of nonlinear fuzzy Volterra integral equa- tions(NFVIEs) using successive approximations scheme and block-pulse functions. Additionally, the convergence analysis of the presented ap- proach is investigated involving Lipschitz and several conditions and error bound between the approximate and the exact solution is pro- vided. Finally, to approve the outcomes concerned with the theory a numerical experiment is considered.

Discrete collocation method for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations

This paper proposes a discrete collocation method based on the fuzzy Lagrange interpolation and the fuzzy Gauss-Legendre quadrature formula for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations. Firstly, the existence and uniqueness of the solution of the original equation are proved by using a two-dimensional Gronwall inequality and an iterative method. Secondly, the discrete collocation method is utilized to convert the linear and nonlinear fuzzy Volterra integral equations into the corresponding linear and nonlinear systems of algebraic equations. Finally, the error estimation and convergence analysis of the proposed numerical method are discussed in terms of the uniform modulus of continuity. Some numerical experiments illustrate that the scheme possesses a higher accuracy and yields a better approximation when compared with other known methods.

On the Smoothness of the Solution of Fuzzy Volterra Integral Equations of the Second Kind with Weakly Singular Kernels

In this paper. we consider fuzzy Volterra integral equation of the second kind with weakly singular kernel. We prove existence and uniqueness of the solution. When analyzing the convergence of a numerical method for a given integral equation one needs information about the smoothness of the exact solution. We describe the smoothness of the solution, assuming that the sign of the kernel can change only along the horizontal and vertical lines.

Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth Order Runge-Kutta Method

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

An efficient method to solve fuzzy Volterra integral equations using Fibonacci polynomials

According to a huge interest in implementation of the fuzzy Volterra integral equations, especially the second kind, researchers have been investigating to solve such equations using numerical methods since analytical ones might not be accessible usually. In this research paper, we introduce a new approach based on Fibonacci polynomials collocation method to numerically solve them. Several properties of such polynomials were considered to implement in the collocation method due to approximate the solution of the second kind of fuzzy Volterra integral equations. We approved the existence, uniqueness of the solution, convergence and the error analysis of the proposed method in detail. In order to show the authenticity and applicability of the proposed method, we employed several illustrative examples. The numerical results show that the convergence and precision of the recent method were in a good settlement with the exact solution. Also, the calculations of the suggested method are simple and low computational complexity in respect to other methods as an advantage feature of the presented approach.

Efficiency of iterative filtering method for solving Volterra fuzzy integral equations with a delay and material investigation

In this article, we analyse the convergence of an iterative method called Adomian decomposition method to solve the Fuzzy Volterra Integral Equations (FVIE) with time lag, the uniqueness and convergence of the method are equivalent. The concentration of the method has been proven and the theoretical results obtained have been tested with a numerical example.AMS2010 Mathematics Subject Classification: 34A07, 34A12, 78M30.

Solution and Analysis of the Fuzzy Volterra Integral Equations via Homotopy Analysis Method

Homotopy Analysis Method (HAM) is semi-analytic method to solve the linear and nonlinear mathematical models which can be used to obtain the approximate solution. The HAM includes an auxiliary parameter, which is an efficient way to examine and analyze the accuracy of linear and nonlinear problems. The main aim of this work is to explore the approximate solutions of fuzzy Volterra integral equations (both linear and nonlinear) with a separable kernel via HAM. This method provides a reliable way to ensure the convergence of the approximation series. A new general form of HAM is presented and analyzed in the fuzzy domain. A qualitative convergence analysis based on the graphical method of a fuzzy HAM is discussed. The solutions sought by the proposed method show that the HAM is easy to implement and computationally quite attractive. Some solutions of fuzzy second kind Volterra integral equations are solved as numerical examples to show the potential of the method. The results also show that HAM provides an easy way to control and modify the convergence area in order to obtain accurate solutions.

A hybrid method for solving fuzzy Volterra integral equations of separable type kernels

The paper deals with the computation of solutions of fuzzy Volterra integral equations with degenerate kernel by applying a hybrid method. The proposed method is built on Laplace transform coupled with Adomian Decomposition Method; Laplace Adomian Decomposition Method abbreviated as (LADM). In the considered equation the unknown function has a solution in terms of infinite series expansion and hence LADM becomes more accurate to give the exact solution. Firstly, using the fuzzy number in term of parametric form, the fuzzy Volterra integral equation is converted to two crisp integral equations and then LADM is applied to get the exact fuzzy solutions of fuzzy linear Volterra integral equations. For the illustration, some examples of considered equations are solved to highlight the robustness, efficiency and the applicability of the developed scheme. The obtained results play an important role in developing the theory of fuzzy analytical dynamic equations.

Iterative fuzzy Bernstein polynomials method for nonlinear fuzzy Volterra integral equations

An iterative method based on fuzzy Bernstein polynomials is presented for solving nonlinear fuzzy Volterra integral equations. To prove the convergence of the method, an error estimate is given in terms of Lipschitz constants. The accuracy of this method is illustrated by some numerical experiments that confirm the convergence stated in the theoretical result.