Fuzzy Volterra Integro-differential Equations Research Articles

A Comparison of Fuzzy Membership Numbers on the Convergence of Numerical Approaches to Solving Integro-Differential Equations

In this article examines the convergences stability of Variational Iteration Method (VIM) for solving Fuzzy Volterra Integro-Differential Equations (FVIDE) of second kind under the Seikkala derivative [13]. The advantage of the proposed method in this study compared with Adomian decomposition Method (ADM). The undefined variables are represented by membership values in trapezoids and triangles. The results of the two methods are compared to demonstrate the efficacy of fuzzy numbers in terms of increasing membership values.

Solving Fuzzy Volterra Integro-Differential Equations By Using Fuzzy Kamal Transform

In this paper, the fuzzy Kamal transform is used to solve fuzzy Volterra integral-differential equations, which is based on Kamal transform. Kamal transform takes very little computation and time. Numerical examples are given to prove the effectiveness of Kamal transform in solving fuzzy Volterra integro-differential equations

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>

Numerical Simulation of Fuzzy Volterra Integro-differential Equation using Improved Runge-Kutta Method

In this research, fourth-order Improved Runge-Kutta method with three stages for solving fuzzy Volterra integro-differential (FVID) equations of the second kind under the concept of generalized Hukuhara differentiability is proposed. The advantage of the proposed method in this study compared with the same order classic Runge-Kutta method is, Improved Runge-Kutta (IRK) method uses a fewer number of stages in each step which causes less computational cost in total. Here, the integral part is approximated by applying the combination of Lagrange interpolation polynomials and Simpson’s rule. The numerical results are compared with some existing methods such as the fourth-order Runge-Kutta (RK) method, variational iteration method (VIM), and homotopy perturbation method (HPM) to prove the efficiency of IRK method. Based on the obtained results, it is clear that the fourth-order Improved Runge-Kutta method with higher accuracy and less number of stages which leads the less computational cost is more efficient than other existing methods for solving FVID equations.

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.

Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

Numerical Investigation to Fuzzy Volterra Integro-Differential Equations via Residual Power Series Method

In this paper, a study of a numerical approximate solution to fuzzy Volterra integro-differential equations is presented under strongly generalised differentiability by applying an influent effective technique, called the Residual Power Series (RPS) method. The solution approach can be expressed on Taylor's series formula in terms of elementary σ-level representation, whereas the coefficients can be computed by utilising its residual functions. Furthermore, a numerical computational example is given to test and validate the proposed method. The results reached show several features concerning the RPS method such as potentiality, generality and superiority to handle many problems arising in physics and engineering.

Soft computing technique for a system of fuzzy Volterra integro-differential equations in a Hilbert space

In this article, we implement a relatively new computational technique, a reproducing kernel Hilbert space method, for solving a system of fuzzy Volterra integro-differential equations in the Hilbert space ⊕j=1n(W22[a,b]⊕W22[a,b]). Based on the concept of the reproducing kernel function combined with Gram-Schmidt orthogonalization process, we represent an exact solution in a form of Fourier series in the reproducing kernel Hilbert space ⊕j=1n(W22[a,b]⊕W22[a,b]). Accordingly, the approximate solution of the system of fuzzy Volterra integro-differential equations is obtained by the n-term intercept of the exact solution and proved to converge to the exact solution. Finally, two numerical examples are presented to illustrate the reliability, appropriateness and efficiency of the 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 reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations

In this article, we propose a new method that determines an efficient numerical procedure for solving second-order fuzzy Volterra integro-differential equations in a Hilbert space. This method illustrates the ability of the reproducing kernel concept of the Hilbert space to approximate the solutions of second-order fuzzy Volterra integro-differential equations. Additionally, we discuss and derive the exact and approximate solutions in the form of Fourier series with effortlessly computable terms in the reproducing kernel Hilbert space W_{2}^{3} [ a,b ] oplus W_{2}^{.3} [ a,b ]. The convergence of the method is proven and its exactness is illustrated by three numerical examples.

Generalization of Seikkala derivative and differential transform method for fuzzy Volterra integro-differential equations

Generalization of Seikkala derivative and differential transform method for fuzzy Volterra integro-differential equations

Solving Fuzzy Volterra Integrodifferential Equations of Fractional Order by Bernoulli Wavelet Method

A matrix method called the Bernoulli wavelet method is presented for numerically solving the fuzzy fractional integrodifferential equations. Using the collocation points, this method transforms the fuzzy fractional integrodifferential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown coefficients. To illustrate the method, it is applied to certain fuzzy fractional integrodifferential equations, and the results are compared.

Application of Homotopy Analysis Method to Solve Fuzzy Volterra Integro-Differential Equations

Application of Homotopy Analysis Method to Solve Fuzzy Volterra Integro-Differential Equations

Approximating the Solution of the Linear and Nonlinear Fuzzy Volterra Integrodifferential Equations Using Expansion Method

The present research study introduces an innovative method applying power series to solve numerically the linear and nonlinear fuzzy integrodifferential equation systems. Finally, it ends with some examples supporting the idea.

Homotopy Analysis Method for Solving Fuzzy Integro-Differential Equations

In this paper, the numerical solution of the linear and nonlinear fuzzy Volterraintegro-differential equations have been investigated using the analytical method namely homotopy analysis method (HAM) and then the proposed method is illustrated by solving two numerical examples. It was found that the HAM provides a simple way to adjust and control the convergence region of solution series by introducing a nonzero auxiliary parameter h.

Numerical solution of linear fuzzy Volterra integro-differential equations by variational iteration method

In this paper, we focus on linear fuzzy Volterra integro-differential equations and propose a new method, namely “variational iteration method VIM” for numerical solving them. Also, we provide the convergence of VIM when it is applied to system of linear Volterra integro-differential equations. Finally, using the proposed method, we give some illustrative examples and compare the solutions obtained by VIM with those obtained by homotopy perturbation method HPM. The results reveal that the VIM is more powerful than the HPM.

Toward the existence and uniqueness of solutions of second-order fuzzy volterra integro-differential equations with fuzzy kernel

In this paper, we study existence and uniqueness of solutions of second-order fuzzy integro-differential equa- tions with fuzzy kernel under strongly generalized differen- tiability. To this end, four cases are considered to show the existence of the fuzzy solution mentioned equation. Some theorems are proved, and the uniqueness of the fuzzy solution is discussed step by step. Finally, the illustrated examples are solved to investigate the conditions of theorems.

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.