Parametric Form Of Fuzzy Numbers Research Articles

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of 0≤ ξ ≤ 1. The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example.

Numerical Solution for Fuzzy Time-Fractional Cancer Tumor Model with a Time-Dependent Net Killing Rate of Cancer Cells.

A cancer tumor model is an important tool for studying the behavior of various cancer tumors. Recently, many fuzzy time-fractional diffusion equations have been employed to describe cancer tumor models in fuzzy conditions. In this paper, an explicit finite difference method has been developed and applied to solve a fuzzy time-fractional cancer tumor model. The impact of using the fuzzy time-fractional derivative has been examined under the double parametric form of fuzzy numbers rather than using classical time derivatives in fuzzy cancer tumor models. In addition, the stability of the proposed model has been investigated by applying the Fourier method, where the net killing rate of the cancer cells is only time-dependent, and the time-fractional derivative is Caputo's derivative. Moreover, certain numerical experiments are discussed to examine the feasibility of the new approach and to check the related aspects. Over and above, certain needs in studying the fuzzy fractional cancer tumor model are detected to provide a better comprehensive understanding of the behavior of the tumor by utilizing several fuzzy cases on the initial conditions of the proposed model.

Ranking parametric form of fuzzy numbers by defuzzification based on centroids value and ambiguity

This paper proposes a new method to rank the parametric form of fuzzy numbers based on defuzzification. The defuzzification process use centroids, value, ambiguity and decision levels on fuzzy number developed from the parametric form of a generalized fuzzy number. The proposed method avoids reducing function to remove lower alpha levels and can overcome the shortcomings in some of the existing fuzzy ranking methods. The proposed method can effectively rank symmetric fuzzy numbers with the same core and different heights, fuzzy numbers with the same support and different cores, crisp numbers, crisp numbers having the same support and different heights, and fuzzy numbers having compensation of areas. A demonstration of the proposed method through examples and a comparative study with other methods in the literature shows that the proposed method gives effective results.

Imprecisely defined fractional-order Fokker–Planck equation subjected to fuzzy uncertainty

Fokker–Planck equation with interval and fuzzy uncertainty has been considered in this paper. Also the derivatives involved with respect to time and space are assumed to be fractional in nature. This problem has been solved using variational iteration method (VIM) along with the double parametric form of fuzzy numbers. For the analysis, both triangular and Gaussian normalised fuzzy sets are taken into consideration. Numerical results for different cases have been obtained and those are depicted in terms of plots and are also compared in special cases for the validation. Moreover, using an important method known as successive approximation method, it has also been verified that the obtained solutions are the same as that of VIM as both methods are equivalent.

Double Parametric Fuzzy Numbers Approximate Scheme for Solving One-Dimensional Fuzzy Heat-Like and Wave-Like Equations

This article discusses an approximate scheme for solving one-dimensional heat-like and wave-like equations in fuzzy environment based on the homotopy perturbation method (HPM). The concept of topology in homotopy is used to create a convergent series solution of the fuzzy equations. The objective of the study is to formulate the double parametric fuzzy HPM to obtain approximate solutions of fuzzy heat-like and fuzzy wave-like equations. The fuzzification and the defuzzification analysis for the double parametric form of fuzzy numbers of the fuzzy heat-like and the fuzzy wave-like equations is carried out. The proof of convergence of the solution under the developed approximate scheme is provided. The effectiveness of the proposed method is tested by numerically solving examples of fuzzy heat-like and wave-like equations where results indicate that the approach is efficient not only in terms of accuracy but also with respect to CPU time consumption.

Analysis of time‐fractional fuzzy vibration equation of large membranes using double parametric based Residual power series method

AbstractThe study of the time‐fractional vibration equation (VE) of large membranes is vital due to its widespread applications. Various researchers have investigated the titled problem in which the variables and parameters have been considered as crisp/exact. However, in actual practice, this may contain uncertainty because of errors in observations, maintenance induced error, and other errors. In this investigation, we have considered these uncertainties as interval/fuzzy, and a method, namely double parametric form of fuzzy numbers are used to solve the uncertain fractional vibration model of order . The titled problem has been solved under various cases using the Residual power series method (RPSM). The performance of the present method is well in terms of computational efficiency. The consistency of the method is displayed by providing numerical solutions for various cases. Obtained results are depicted in terms of figures and are also compared with individual cases.

Homotopy Perturbation Method for Solving Linear Fuzzy Delay Differential Equations Using Double Parametric Approach

Delay differential equations (known as DDEs) are a broad use of many scientific researches and engineering applications. They come because the pace of the shift in their mathematical models relies all the basis not just on their present condition, but also on a certain past cases. In this work, we propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using the Homotopy Perturbation Method with double parametric form fuzzy numbers. The detailed algorithm of the approach to fuzzification and defuzzificationis analysis is provided. In the initial conditions of the proposed problem there are uncertainties with regard to the triangular fuzzy number. A double parametric form of fuzzy numbers is defined and applied for the first time in this topic for the present analysis. This method's simplicity and ability to overcome delay differential equations without complicating Adomian polynomials or incorrect nonlinear assumptions. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show the features of this proposed method, a numerical example is illustrated, involving first order fuzzy delay differential equation. These findings indicate that the suggested approach is very successful and simple to implement.

Numerical solutions of fuzzy time fractional advection‐diffusion equations in double parametric form of fuzzy number

Fractional partial differential equations are a generalization of classical partial differential equations which can, in certain circumstances, give a better description of certain phenomena. In this paper, two implicit finite difference schemes are developed, analyzed, and applied to solve an initial boundary value problem involving fuzzy time fractional advection‐diffusion equation with fractional order 0<α≤1. The fuzziness of the problem considered appears in the initial and boundary conditions. A computational mechanism is presented based on double parametric form of fuzzy number to transfer the problem from uncertain to crisp form. The stability of the proposed schemes is analyzed by means of the Von Neumann method and were found to be unconditionally stable. The scheme was applied to an example to illustrate the feasibility.

A compact Crank–Nicholson scheme for the numerical solution of fuzzy time fractional diffusion equations

Fuzzy fractional partial differential equations are a generalization of classical fuzzy partial differential equation which can, in certain circumstances, provide a better explanation for certain phenomena. In this paper, a compact Crank–Nicholson scheme is developed, analyzed and applied to solve the fuzzy time diffusion equation with fractional order $$0 < \alpha \le 1$$ . It is based on double parametric form of fuzzy numbers and the time fractional derivative is defined using the Caputo definition. The truncation error and stability of the proposed scheme is discussed. The compact Crank–Nicholson scheme is shown to be a feasible scheme via a numerical example.

Uncertain dynamic responses of imprecisely defined arbitrary order fractionally damped beam subject to various loads

PurposeRecently, fractional differential equations have been used to model various physical and engineering problems. One may need a reliable and efficient numerical technique for the solution of these types of differential equations, as sometimes it is not easy to get the analytical solution. However, in general, in the existing investigations, involved parameters and variables are defined exactly, whereas in actual practice it may contain uncertainty because of error in observations, maintenance induced error, etc. Therefore, the purpose of this paper is to find the dynamic response of fractionally damped beam approximately under fuzzy and interval uncertainty.Design/methodology/approachHere, a semi analytical approach, variational iteration method (VIM), has been considered for the solution. A newly developed form of fuzzy numbers known as double parametric form has been applied to model the uncertainty involved in the system parameters and variables.FindingsVIM has been successfully implemented along with double parametric form of fuzzy number to find the uncertain dynamic responses of the fractionally damped beam. The advantage of this approach is that the solution can be written in power series or compact form. Also, this method converges rapidly to have the accurate solution. The uncertain responses subject to impulse and step loads have also been computed and the behaviours of the responses are analysed. Applying the double parametric form, it reduces the computational cost without separating the fuzzy equation into coupled differential equations as done in traditional approaches.Originality/valueUncertain dynamic responses of fuzzy fractionally damped beam using the newly developed double parametric form of fuzzy numbers subject to unit step and impulse loads have been obtained. Gaussian fuzzy numbers are used to model the uncertainties. In the methodology using the alpha cut form, corresponding beam equation is first converted to an interval-based fuzzy equation. Next, it has been transformed to crisp form by applying double parametric form of fuzzy numbers. Finally, VIM has been applied to solve the same for the general fuzzy responses. Various numerical examples have been taken in to consideration.

Solving imprecisely defined vibration equation of large membranes

PurposeVibration of large membranes has great utility in engineering application such as in important parts of drums, pumps, microphones, telephones and other devices. So, to obtain a numerical solution of this type of problems is necessary and important. In general, in existing approaches, involved parameters and variables are defined exactly. Whereas in actual practice, it may contain uncertainty owing to error in observations, maintenance-induced error, etc. So, the main purpose of this paper is to solve this important problem numerically under fuzzy and interval uncertainty to have an uncertain solution and to study its behaviour.Design/methodology/approachIn this study, the authors have considered a new approach is known as double parametric form of fuzzy number to model uncertain parameters. Along with this a semianalytical approach, i.e. variational iteration method, has been used to obtain uncertain bounds of the solution.FindingsThe variational iteration method has been successfully implemented along with the double parametric form of fuzzy number to find the uncertain solution of the vibration equation of a large membrane. The advantage of this approach is that the solution can be written in a power series or a compact form. Also, this method converges rapidly to obtain an accurate solution. Various cases depending on the functional value involved in the initial conditions have been studied and the behaviour has been analysed. Applying the double parametric form reduces the computational cost without separating the fuzzy equation into coupled differential equations as done in traditional approaches.Originality/valueThe vibration equation of large membranes has been solved under fuzzy and interval uncertainty. Uncertainties have been considered in the initial conditions. New approaches, i.e. variational iteration method along with the double parametric form, have been applied to solve the vibration equation of large membranes.

Iterative method for approximate solution of fuzzy integro-differential equations

In this paper, we interpret a nonlinear fuzzy Fredholm integro-differential equations by using the strongly generalized differentiability concept. Based on the parametric form of a fuzzy number, a fuzzy integro-differential equation converts to two systems of integro-differential equations in the crisp case. Also, we use the parametric form of fuzzy number, and an iterative approach for obtaining approximate solution for a class of nonlinear fuzzy Fredholm integro-differential equation of the second kind is proposed. This paper presents a method based on Newton–Cotes methods with positive coefficient. Then we obtain approximate solution of the nonlinear fuzzy integro-differential equations by an iterative approach.

A new method of ranking parametric form of fuzzy numbers using value and ambiguity

Most of the real-life, medical and engineering problem may often involve decision-making under uncertain circumstances. These problems may be interpreted in terms of fuzzy numbers. Hence, a proper method of comparing and ranking these fuzzy numbers is very essential. In recent years, numerous way of ranking fuzzy numbers have been proposed. Most of the existing approaches have shortcomings and limitations under certain circumstances. In particular symmetric fuzzy numbers having same core and different supports, fuzzy numbers having same support and different cores, crisp valued fuzzy numbers, crisp valued fuzzy numbers having same support and different heights and fuzzy numbers having compensation of areas are not properly ranked in most of the studies. Hence, a new method of ranking fuzzy number using the value and the ambiguity, which are ill-defined magnitude and amount of vagueness present in the ill-defined magnitude respectively, at different decision level has been proposed. The proposed method has been discussed on the concept of parametric form of fuzzy number with the defuzzifiers are at different heights and the concept of decision levels. This concept of parametric form of fuzzy number with the defuzzifiers are at different heights has been discussed for the first time. The proposed method has been compared with other existing methods through numerical examples and found to outperform in all situations. The results demonstrate that the proposed method can clearly overcome the addressed limitations and shortcomings. As a possible application, a problem on fuzzy linear programming has been addressed.

NUMERICAL SOLUTIONS OF NONLINEAR FUZZY FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS OF THE SECOND KIND

In this paper, we use parametric form of fuzzy number, then aniterative approach for obtaining approximate solution for a classof nonlinear fuzzy Fredholmintegro-differential equation of the second kindis proposed. This paper presents a method based on Newton-Cotesmethods with positive coefficient. Then we obtain approximatesolution of the nonlinear fuzzy integro-differential equations by an iterativeapproach.

Minimal solution of fuzzy linear systems

In this paper, we use parametric form of fuzzy number and we converta fuzzy linear system to two linear system in crisp case. Conditions for the existence of a minimal solution to $mtimes n$ fuzzy linear equation systems are derived and a numerical procedure for calculating the minimal solution is designed. Numerical examples are presented to illustrate the proposed method.

New approach to solve fully fuzzy system of linear equations using single and double parametric form of fuzzy numbers

This paper proposes two new methods to solve fully fuzzy system of linear equations. The fuzzy system has been converted to a crisp system of linear equations by using single and double parametric form of fuzzy numbers to obtain the non-negative solution. Double parametric form of fuzzy numbers is defined and applied for the first time in this paper for the present analysis. Using single parametric form, the n × n fully fuzzy system of linear equations have been converted to a 2n × 2n crisp system of linear equations. On the other hand, double parametric form of fuzzy numbers converts the n × n fully fuzzy system of linear equations to a crisp system of same order. Triangular and trapezoidal convex normalized fuzzy sets are used for the present analysis. Known example problems are solved to illustrate the efficacy and reliability of the proposed methods.

Uncertain vibration equation of large membranes

The study of the vibration of large membranes is important due to its well-known applications. There exist various investigations for the above problem where the variables and parameters are given as crisp/exact. In practice, we may not have these parameters exactly but those may be known in some uncertain form. In the present paper, these uncertainties are taken as interval/fuzzy and the authors propose here a new method viz. that of the double parametric form of fuzzy numbers to handle the uncertain problem of large membranes. Finally, the problem has been solved using the Homotopy Perturbation Method (HPM). The present method performs very well in terms of computational efficiency. The reliability of the method is shown for obtaining an approximate numerical solution for different cases. Results are given in terms of plots and are also compared in special cases.

Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations

The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 < α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.

Dynamic response of imprecisely defined beam subject to various loads using Adomian decomposition method

Present paper proposes a new technique based on double parametric form of fuzzy numbers to solve an uncertain beam equation using Adomian decomposition method subject to unit step and impulse loads. Uncertainties appear in the initial conditions are considered in terms of triangular convex normalized fuzzy sets. Using the single parametric form viz. α-cut form of fuzzy numbers, the fuzzy beam equation is converted first to an interval based fuzzy differential equation. Next this differential equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same is solved by Adomian decomposition method symbolically to obtain the uncertain bounds of the dynamic response. Obtained results are depicted in term of plots to show the efficiency and powerfulness of the present analysis.

NUMERICAL SOLUTIONS OF FUZZY NONLINEAR INTEGRAL EQUATIONS OF THE SECOND KIND

In this paper, we use the parametric form of fuzzy numbers, and aniterative approach for obtaining approximate solution for a classof fuzzy nonlinear Fredholm integral equations of the second kindis proposed. This paper presents a method based on Newton-Cotesmethods with positive coefficient. Then we obtain approximatesolution of the fuzzy nonlinear integral equations by an iterativeapproach.