Abstract
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.
Highlights
Modeling processes of natural phenomena create perceptions and general impressions about the dynamic behavior of any physical system that may involve uncertain parameters that result from many factors such as measurement errors, estimates, expectations, and deficient data
A modified numerical algorithm has been profitably designed in light of reproducing Kernel Hilbert space (RKHS) method and employed to get approximate solutions of fuzzy fractional integrodifferential equations by means of Atangana-Baleanu-Caputo gHdifferentiability
Characterization theorem was established for ABC-fractional order, in which the studied fuzzy fractional model was transformed into a crisp system of fractional IVPs under fuzzy ABC calculus
Summary
Modeling processes of natural phenomena create perceptions and general impressions about the dynamic behavior of any physical system that may involve uncertain parameters that result from many factors such as measurement errors, estimates, expectations, and deficient data. The most common are the concepts of Riemann-Liouville and Caputo which bring some privacy They involve a singular kernel function that may adversely affect a realistic understanding of real-world problems. The authors [4, 56] defined a novel operator of fractional derivatives based on Atangana-Baleanu-Caputo (ABC) in view of fuzzy valued function with form of parametric interval, called. We intend to study the effect of ABC gH-differentiability on the solution of different types of fuzzy fractional integrodifferential equations (FFIDEs). Motivated by the aforementioned discussion, this numerical research aims to design a novel iterative algorithm to obtain solutions to fuzzy integrodifferential equations in terms of the new ABC-fractional concept containing nonsingular and nonlocal kernel under gH-differentiability in addition to studying the effect of ABC-fractional derivative on these solutions.
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