Fuzzy Solutions to Partial Differential Equations: Adaptive Approach

Chen, Yung-Yue; Chang, Young Chul; Chen, Bor‐Sen

doi:10.1109/tfuzz.2008.2005010

Cited by 56 publications

(7 citation statements)

References 18 publications

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“…Thus, in the same manner, the double parametric fuzzy HPM converges to the exact solution of Equation (31) for each r, b ∈ [0, 1]. In this example, the error analysis of double parametric HPM series solution of Equation (31) for all r, b ∈ [0, 1] is defined by the absolute errors between the exact solution(34) and the approximate solution(28) in the following form: E(x, t, r, b) 5,10 = Ũ (x, t, r, b) −ũ(x, t, r, b) 5,10 . (37)…”

mentioning

confidence: 94%

“…Therefore, the aims of the paper is to develop an approximate-analytical scheme based on the double parametric fuzzy HPM to obtain approximate solutions of one-dimensional fuzzy heat-like and fuzzy wave-like equations in the form of double-parametric fuzzy numbers and also to prove the convergence of the obtained solutions. We wish to point out that the emphasis in this paper is on the computational and formal issues; a discussion on the physical meaning of the fuzzy solutions to partial differential equations can be retrieved in [28].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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.

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confidence: 94%

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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“…Therefore, the aim of this work is to reformulate the standard finite difference methods to obtain new forms of the cantered time centre space and implicit schemes and prove the stability thermos for the numerical solution of the FWE under the double parametric form of fuzzy numbers for the first time. We would also like to point out that this paper focuses on computational and formal issues; discussion can also be retrieved in [20] on the physical significance of fading solutions to partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…It should be noted that some of the fuzzy sets concepts and definitions not given in this work are standard. Notions of fuzzy level sets, fuzzy numbers, fuzzy functions, fuzzy Zadeh expansion theory and fuzzy H-derivatives can be checked from the literature, e.g., [20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form

et al. 2021

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The use of fuzzy partial differential equations has become an important tool in which uncertainty or vagueness exists to model real-life problems. In this article, two numerical techniques based on finite difference schemes that are the centered time center space and implicit schemes to solve fuzzy wave equations were used. The core of the article is to formulate a new form of centered time center space and implicit schemes to obtain numerical solutions fuzzy wave equations in the double parametric fuzzy number approach. Convex normalized triangular fuzzy numbers are represented by fuzziness, based on a double parametric fuzzy number form. The properties of fuzzy set theory are used for the fuzzy analysis and formulation of the proposed numerical schemes followed by the new proof stability thermos under in the double parametric form of fuzzy numbers approach. The consistency and the convergence of the proposed scheme are discussed. Two test examples are carried out to illustrate the feasibility of the numerical schemes and the new results are displayed in the forms of tables and figures where the results show that the schemes have not only been effective for accuracy but also for reducing computational cost.

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“…The fuzzy partial differential equations FPDE are very important in mathematical models of physical, chemical, biological, economics and other fields. Some mathematicians have studied solution of FPDE by numerical methods (Afshar Kermani and Saburi 2007; Allahviranloo 2002; Barkhordari Ahmadi and Kiani 2011; Buckley and Feuring 1999; Chen et al 2009; Farajzadeh et al 2010; Moghadam and Jalal 2011; Rouhparvar et al 2010; Verma et al 2009). In this work, we present the Picard method to solve the Cauchy reaction-diffusion equation as follows: …”

Section: Introductionmentioning

confidence: 99%

Solving Cauchy reaction-diffusion equation by using Picard method

Behzadi

2013

SpringerPlus

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In this paper, Picard method is proposed to solve the Cauchy reaction-diffusion equation with fuzzy initial condition under generalized H-differentiability. The existence and uniqueness of the solution and convergence of the proposed method are proved in details. Some examples are investigated to verify convergence results and to illustrate the efficiently of the method. Also, we obtain the switching points in examples.

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Fuzzy Solutions to Partial Differential Equations: Adaptive Approach

Cited by 56 publications

References 18 publications

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

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

Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form

Solving Cauchy reaction-diffusion equation by using Picard method

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