Fuzzy Integro-differential Equations Research Articles

Studies over the existence of a certain impulse-based fuzzy integrodifferential equations of the Sobolev type

The present investigation employs impulses and a non-local constraint to prove the existence are some various types of abstract differential and integrodifferential equations related to the Sobolev type. Semigroup theory, specifically variants of constant formula, is utilized to get the analytical results for those equations. Furthermore, findings using the Banach fixed point approach were examined using fuzzy numbers with values spanning the En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {E}_n$$\\end{document} range, which includes the normal, convex, upper semi-continuous, and compactly supported interval. A description is given for each situation to illustrate the principle.

The Fuzzy Differential Transform Method for the Solution of the System of Fuzzy Integro-Differential Equations Arising in Biological Model

This article deals with the implementation of fuzzy differential transform method for solving a system of nonlinear fuzzy integro-differential equations. This system appears in a model of biological species living together. Though the differential transform method is an iterative method, the current approach reduces this model to a set of nonlinear algebraic equations due to its delay terms. The basic definitions and theorems are first presented. The applicability and accuracy of the current methodologies have been demonstrated through the discussion of a few exemplary situations.

Numerical Approach for Solving fuzzy Integro-Differential Equations

In this paper, we consider a new class of fuzzy functions called Fuzzy Integro- Differential Equations. Some numerical methods, such as Euler, have been used to determine the solutions of these equations. We extend these numerical techniques to find the optimal solutions by using control parameters, the extended difference Euler technique is used for this. Based on the parametric form of the fuzzy number, the Integro- Differential Equation is divided into two systems of the second kind. Illustrative examples are given to demonstrate the high precision and good performance of the new class. Graphical representations reveal the symmetry between lower and upper-cut represent of fuzzy solutions and may be helpful for a better understanding of fuzzy models in artificial intelligence and medical science. The results show that the extended Euler method is more accurate in terms of absolute error.

Fixed point results of fuzzy mappings with applications

<abstract><p>Jleli and Samet introduced the notion of $ \mathcal{F} $-metric space as a generalization of traditional metric space and proved Banach contraction principle in the setting of these generalized metric spaces. The aim of this article is to utilize $ \mathcal{F} $-metric space and establish some common $ \alpha $-fuzzy fixed point theorems for rational ($ \beta $-$ \phi) $-contractive conditions. Our results extend, generalize and unify some well-known results in the literature. As application of our main result, we discuss the solution of fuzzy integrodifferential equations in the setting of a generalized Hukuhara derivative.</p></abstract>

A modified reproducing Kernel Hilbert space method for solving fuzzy fractional integro-differential equations

The aim of this paper is to extend the application of the reproducing kernel Hilbert space method (RKHSM) to solve linear and non-linear fuzzy integro-differential equations of fractional order under Caputo's H-differentiability. The analytic and approximate solutions are given in series form in term of their parametric form in the space $W_2^2 [a,b] \bigoplus W_2^2 [a,b]$. Several examples are carried out to show the effectiveness and the absence of complexity of the proposed method

Existence and uniqueness of intuitionistic fuzzy solution for semilinear intuitionistic fuzzy integro-differential equations with nonlocal conditions

n this paper, we study the existence and uniqueness of an intuitionistic fuzzy solution for semi-linear intuitionistic fuzzy integro-differential equations with non-local conditions using the Banach fixed point theorem. Theorem on the existence and uniqueness of intuitionistic fuzzy solution for these problems with nonlocal conditions are presented under certain assumptions. Finally, an example is established to illustrate the effectiveness of this theorem.

A Quantitative Approach to $$n{\text {th}}$$-Order Nonlinear Fuzzy Integro-Differential Equation

A Quantitative Approach to $$n{\text {th}}$$-Order Nonlinear Fuzzy Integro-Differential Equation

Effective Approach to Construct Series Solutions for Uncertain Fractional Differential Equations

Purpose: We construct the analytical approximate resiual power fuzzy series solutions of fuzzy conformable fractional differential equations in an -level depiction in the sense of strongly generalized -fuzzy conformable derivative in which of the all initial conditions are taken to be fuzzy numbers. Methodology: The certain fuzzy conformable fractional differential equation under strongly generalized -fuzzy derivative is converted to a crisp one as a family of differential inclusions and solved via resiual power method. The main drawback concerning the use of differential inclusions is that it does not contain a fuzzification of the differential operator; instead, the solution is not essentially a fuzzy valued function. Findings: (i) To show the efficiency of our proposed method: Several important and attractive test examples, which included the fractional conformable fuzzy integro-differential equation are discussed and solved in detail. (ii) To show the stability of approximate solutions to specific problems: some graphical results, numerical comparisons and tabulate data are created and discussed at different values of Value: Using the residual power series analysis methos is a powerful and easy-to-use analytic tool to solve initial problems on fuzzy conformable fractional differential equations and it successfully applied to solve real life problems such as the inductance–resistance–capacitance, RLC-series circuit.

Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method

<abstract> <p>In this paper, an efficient technique called Optimal Homotopy Asymptotic Method has been extended for the first time to the solution of the system of fuzzy integro-differential equations of fractional order. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The fuzzy fractional derivatives are defined in Caputo sense. It is followed by suggesting a new result from Optimal Homotopy Asymptotic Method for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of system of fuzzy integro-differential equations of fractional order and finally, we demonstrate a numerical example. The validity and efficiency of the proposed technique are demonstrated via these numerical examples which depend upon the parametric form of the fuzzy number. The optimum values of convergence control parameters are calculated using the well-known method of least squares, obtained results are compared with fractional residual power series method. It is observed from the results that the suggested method is accurate, straightforward and convenient for solving system of fuzzy Volterra integrodifferential equations of fractional order.</p> </abstract>

On solution of fuzzy Volterra integro-differential equations

In this article, the Volterra integro-differential equation of separable type kernels is considered under the fuzzy concept. A hybrid technique is used to solve fuzzy integro-differential equations under the notion of generalized Hukuhara differentiability. To build up a suitable algorithm, Laplace transformation coupled with the Adomian decomposition method (LADM) is used which is an essential and profitable algorithm for setting the fuzzy Volterra integro-differential equations. The proposed method is illustrated by two test problems that show the convergence of the series solution to the exact solution in closed form. The numerical results of the examples are depicted by appropriate graphs at different uncertainty values. Additionally, the graphical portrayals show the fruitfulness and accuracy of the proposed method.

APPLICATION OF AN EFFECTIVE METHOD ON THE SYSTEM OF NONLINEAR FUZZY INTEGRO-DIFFERENTIAL EQUATIONS

In real world physical applications purpose, it is complicated to acquire an exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic and therefore creating the need for the use of reliable and efficient techniques in the solution of fuzzy differential equations. The purpose of this research paper is to utilize the reliable analytic approach of homotopy perturbation Sumudu transform method for better understanding of systems of non-linear fuzzy integro-differential equations, while using the concept of fuzzy parameter in certain dynamical problems to remove the hurdles faced in numerical approach. These mathematical models are of great interest in engineering and physics. Some numerical examples are also given to demonstrate the efficiency and superiority of the method, followed by graphical representation of the comparison of exact and approximated solution by using Maple 2017

Study of Nonlinear Fuzzy Integro-differential Equations Using Mathematical Methods and Applications

Study of Nonlinear Fuzzy Integro-differential Equations Using Mathematical Methods and Applications

Application and comparison of the two efficient algorithms to solve the pantograph Volterra fuzzy integro-differential equation

In this paper, collocation method based on sinc function and Bernoulli wavelet is proposed to find numerical solution of pantograph Volterra fuzzy integro-differential equation. In the first step, the fuzzy integro-differential equation is converted to a system of integro-differential equations in crisp case. Then by using the proposed algorithms, this system is reduced to a system of algebraic equations which is solved by Newton’s method. Also convergence analysis of the methods is given. We illustrate the accuracy of these algorithms by two examples and compare the obtaining results and the computational implementations.

On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space

Many dynamic systems can be modeled by fractional differential equations in which some external parameters occur under uncertainty. Although these parameters increase the complexity, they present more acceptable solutions. With the aid of Atangana-Baleanu-Caputo (ABC) fractional differential operator, an advanced numerical-analysis approach is considered and applied in this work to deal with different classes of fuzzy integrodifferential equations of fractional order fitted with uncertain constraints conditions. The fractional derivative of ABC is adopted under the generalized H-differentiability (g-HD) framework, which uses the Mittag-Leffler function as a nonlocal kernel to better describe the timescale of the fuzzy models. Towards this end, applications of reproducing kernel algorithm are extended to solve classes of linear and nonlinear fuzzy fractional ABC Volterra-Fredholm integrodifferential equations. Based on the characterization theorem, preconditions are established under the Lipschitz condition to characterize the fuzzy solution in a coupled equivalent system of crisp ABC integrodifferential equations. Parametric solutions of the ABC interval are provided in terms of rapidly convergent series in Sobolev spaces. Several examples of fuzzy ABC Volterra-Fredholm models are implemented in light of g-HD to demonstrate the feasibility and efficiency of the designed algorithm. Numerical and graphical representations of both classical Caputo and ABC fractional derivatives are presented to show the effect of the ABC derivative on the parametric solutions of the posed models. The achieved results reveal that the proposed method is systematic and suitable for dealing with the fuzzy fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.

Numerical solutions of fuzzy integro-differential equations of the second kind

Numerical solutions of fuzzy integro-differential equations of the second kind

Double Fuzzy Sumudu Transform to Solve Partial Volterra Fuzzy Integro-Differential Equations

In this paper, the double fuzzy Sumudu transform (DFST) method was used to find the solution to partial Volterra fuzzy integro-differential equations (PVFIDE) with convolution kernel under Hukuhara differentiability. Fundamental results of the double fuzzy Sumudu transform for double fuzzy convolution and fuzzy partial derivatives of the n-th order are provided. By using these results the solution of PVFIDE is constructed. It is shown that DFST method is a simple and reliable approach for solving such equations analytically. Finally, the method is demonstrated with examples to show the capability of the proposed method.

Numerical investigation of the fuzzy integro-differential equations by He$’$s Homotopy perturbation method

Numerical investigation of the fuzzy integro-differential equations by He$’$s Homotopy perturbation method

Sumudu Decomposition Method for Solving Fuzzy Integro-Differential Equations

Different results regarding different integro-differentials are usually not properly generalized, as they often do not satisfy some of the constraints. The field of fuzzy integro-differentials is very rich these days because of their different applications and functions in different physical phenomena. Solutions of linear fuzzy Volterra integro-differential equations (FVIDEs) are more generalized and have better applications. In this report, the Sumudu decomposition method (SDM) was used to find the solution to some linear and nonlinear fuzzy integro-differential equations (FIDEs). Some examples are given to show the validity of the presented method.

Fuzzy Volterra Integro-Differential Equations Using General Linear Method

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods.

Application of Haar Wavelet Method for Solving the Nonlinear Fuzzy Integro-Differential Equations

Haar wavelet method (HWM) is an essential profitable method for settling the nonlinear Fuzzy Fredholm integro-differential equations (NFIDE). The proposed model converts the NFIDE into to nonlinear equations which tackle by the familiar Newton methods. The authors investigate the convergence of this method. Test problems are solved to show the accuracy of our method where the obtained numerical results are compared with Homotopy perturbation method (HPM) and the exact solutions. Graphical portrayals of the correct and obtained estimated arrangements illuminate the exactness of the methodology.