On backward problems for stochastic fractional reaction equations with standard and fractional Brownian motion

The aim of this study is to develop the Fibonacci wavelet method together with the quasi‐linearization technique to solve the fractional‐order logistic growth model. The block‐pulse functions are employed to construct the operational matrices of fractional‐order integration. The fractional derivative is described in the Caputo sense. The present time‐fractional population growth model is converted into a set of nonlinear algebraic equations using the proposed generated matrices. Making use of the quasi‐linearization technique, the underlying equations are then changed to a set of linear equations. Numerical simulations are conducted to show the reliability and use of the suggested approach when contrasted with methods from the existing literature. A comparison of several numerical techniques from the available literature is presented to show the efficacy and correctness of the suggested approach.

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“…with condition 𝜇 s+1 (0) = 𝜇 0 , expand the largest derivative appeared in (32) by Fibonacci wavelet series as…”

Section: Fibonacci Wavelet Methodsmentioning

confidence: 99%

“…To avoid confusion, the fractional derivative will be used throughout the rest of this article in the sense of Caputo. For further studies, we refer previous studies [31][32][33][34][35][36][37][38][39][40].…”

Section: Fractional Calculusmentioning

confidence: 99%

Hybrid Fibonacci wavelet method to solve fractional‐order logistic growth model

Ahmed

Jahan

Nisar

2023

Math Methods in App Sciences

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“…Note that Parseval's equality cannot be applied to our problem with observed data in L p space with p = 2. To overcome these challenges, we learn techniques from the article [30] and in the references [31][32][33][34][35][36][37][38][39][40][41][42][43]. This idea can be summed up as the importance of the technique of using embedding between L p and H s ( ).…”

Section: Our Novelty and Contributionmentioning

confidence: 99%

Fractional evolution equation with Cauchy data in $L^{p}$ spaces

et al. 2022

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In this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in $L^{2}$ L 2 and $H^{s}$ H s . However, there have not been any papers dealing with this problem with observed data in $L^{p}$ L p with $p \neq 2$ p ≠ 2 . We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in $L^{p}$ L p . To our knowledge, $L^{p}$ L p evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.

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“…Some important theoretical results can be found in [1][2][3][4][5][6][7]. Many models that describe many phenomena in the applied sciences can be modeled by fractional differential equations (FDEs); see, for example, [8][9][10][11][12]. Since the analytic solutions for the majority of FDEs do not exist, it is essential to resort to numerical algorithms for handling these types of equations.…”

Section: Introductionmentioning

confidence: 99%

Novel spectral schemes to fractional problems with nonsmooth solutions

Atta

Abd-Elhameed

Moatimid

et al. 2023

Math Methods in App Sciences

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In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin method that reduces each of the FIVP and FPDP into an algebraic system of equations in the unknown expansion coefficients. The class of orthogonal polynomials, namely, Chebyshev polynomials of the fifth kind is utilized. In terms of new basis functions called regular shifted Chebyshev poly‐fractionomials of fifth kind, approximate solutions to the FIVP and FPDP are obtained. Moreover, convergence and error analysis of the two problems are investigated in depth. Some numerical examples are presented with some comparisons. In conclusion, our spectral methods are effective and convenient.

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On backward problems for stochastic fractional reaction equations with standard and fractional Brownian motion

Cited by 20 publications

References 50 publications

Hybrid Fibonacci wavelet method to solve fractional‐order logistic growth model

Hybrid Fibonacci wavelet method to solve fractional‐order logistic growth model

Fractional evolution equation with Cauchy data in $L^{p}$ spaces

Novel spectral schemes to fractional problems with nonsmooth solutions

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