An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations

Alaroud, Mohammad; Al‐Smadi, Mohammed; Ahmad, Rokiah Rozita; Din, Ummul Khair Salma

doi:10.3390/sym11020205

Cited by 47 publications

(30 citation statements)

References 32 publications

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“…For each case, the original fuzzy Duffing equation can be switched to an equivalent crisp system of ODEs. As a result, the proposed method can be used directly to solve the crisp system obtained, without having to be formulated in an uncertain sense [13][14][15][16].…”

Section: Fuzzy Duffing's Equationmentioning

confidence: 99%

“…To beat this uncertainty environment, one may use the concept of fuzziness over coefficients, variables, and initial-boundary conditions instead of crisp ones. So, it is necessary to have some mathematical tools to understand this uncertainty [13][14][15][16]. In this regard, the term "crisp" identifies a formal logic class with indicator function, sometimes called binary-valued logic or standard logic, where the statement is either true or false but not both.…”

Section: Introductionmentioning

confidence: 99%

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Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

et al. 2020

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The mathematical structure of some natural phenomena of nonlinear physical and engineering systems can be described by a combination of fuzzy differential equations that often behave in a way that cannot be fully understood. In this work, an accurate numeric-analytic algorithm is proposed, based upon the use of the residual power series, to investigate the fuzzy approximate solution for a nonlinear fuzzy Duffing oscillator, along with suitable uncertain guesses under strongly generalized differentiability. The proposed approach optimizes the approximate solution by minimizing a residual function to generate r-level representation with a rapidly convergent series solution. The influence, capacity, and feasibility of the method are verified by testing some applications. Level effects of the parameter r are given graphically and quantitatively, showing good agreement between the fuzzy approximate solutions of upper and lower bounds, that together form an almost symmetric triangular structure, that can be determined by central symmetry at r = 1 in a convex region. At this point, the fuzzy number is a convex fuzzy subset of the real line, with a normalized membership function. If this membership function is symmetric, the triangular fuzzy number is called the symmetric triangular fuzzy number. Symmetrical fuzzy estimates of solutions curves indicate a sense of harmony and compatibility around the parameter r = 1. The results are compared numerically with the crisp solutions and those obtained by other existing methods, which illustrate that the suggested method is a convenient and remarkably powerful tool in solving numerous issues arising in physics and engineering.

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Section: Fuzzy Duffing's Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

et al. 2020

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“…According the FRPS method [24][25][26][27], let us assume that the solutions of IVPs (8) and (9) can be written by…”

Section: Mathematical Model Formulationmentioning

confidence: 99%

“…Thus, using the procedures of the RPS algorithm [25][26][27][28], the 4th RPS approximate solution of FBTEs (13) and (14) can be given by ω 4 (t) = 2t 4α Γ(4α+1) . Consequently, the RPS solution at α = 1/2 will be ω(t) = t 2 , which is fully compatible with the exact solution investigated earlier in [32].…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Later, RPSM has been used in generating a fractional power series (FPS) solutions for strongly nonlinear FDEs in the form of a rapidly convergent with a minimum size of calculations without any restrictive hypotheses. Thus, this adaptive can be used as an alternative technique in solving several nonlinear problems arising in engineering and physics [24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

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Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

et al. 2019

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Featured Application: Fractional differential equations play a significant role in modeling certain dynamical systems arising in many fields of applied sciences and engineering. In this paper, the authors develop an attractive analytic-numeric technique, residual power series (RPS) method, for solving fractional Bagley-Torvik equations with a source term involving Caputo fractional derivative. In regard to its simplicity, the method can be applicable to a wide class of fractional partial differential equations, fractional fuzzy differential equations, fractional oscillator equations, and so on.Abstract: Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley-Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

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

Jena

Sedighi

2020

Z Angew Math Mech

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The 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 α(1<α≤2). 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.

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An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations

Cited by 47 publications

References 32 publications

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

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

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