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

Ahmed, Shahid; Jahan, Shah; Nisar, Kottakkaran Sooppy

doi:10.1002/mma.9446

Cited by 15 publications

(3 citation statements)

References 40 publications

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“…Applying the technique of quasi-lineariztion method (QLM) has been successfully proved by many published works, see cf. [39][40][41]. The QLM aimed to not only linearize the model but also keep the accuracy of the proposed algorithm in the same level as the directly employed algorithm.…”

Section: The Qlm-gmbps Collocation Approachmentioning

confidence: 99%

Applications of Modified Bessel Polynomials to Solve a Nonlinear Chaotic Fractional-Order System in the Financial Market: Domain-Splitting Collocation Techniques

Izadi

Srivástava

2023

Computation

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We propose two accurate and efficient spectral collocation techniques based on a (novel) domain-splitting strategy to handle a nonlinear fractional system consisting of three ODEs arising in financial modeling and with chaotic behavior. One of the major numerical difficulties in designing traditional spectral methods is in the handling of model problems on a long computational domain, which usually yields to loss of accuracy. One remedy is to split the underlying domain and apply the spectral method locally in each subdomain rather than on the global domain of interest. To treat the chaotic financial system numerically, we use the generalized version of modified Bessel polynomials (GMBPs) in the collocation matrix approaches along with the domain-splitting strategy. Whereas the first matrix collocation scheme is directly applied to the financial model problem, the second one is a combination of the quasilinearization method and the direct first numerical matrix method. In the former approach, we arrive at nonlinear algebraic matrix equations while the resulting systems are linear in the latter method and can be solved more efficiently. A convergence theorem related to GMBPs is proved and an upper bound for the error is derived. Several simulation outcomes are provided to show the utility and applicability of the presented matrix collocation procedures.

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Section: The Qlm-gmbps Collocation Approachmentioning

confidence: 99%

Applications of Modified Bessel Polynomials to Solve a Nonlinear Chaotic Fractional-Order System in the Financial Market: Domain-Splitting Collocation Techniques

Izadi

Srivástava

2023

Computation

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“…This feature, along with a fast wavelet approach, makes these methods very interesting for analysis and synthesis. Wavelet-based collocation techniques have become more popular in numerical analysis because of their fast convergence, low computational cost, and straightforward procedure [33] . Wavelet techniques are a relatively recent addition to the family of orthogonal functions, with notable and attractive properties such as orthogonality, compact support, unconstrained regularity and good localization.…”

Section: Introductionmentioning

confidence: 99%

On mathematical modelling of measles disease via collocation approach

Ahmed,

Jahan,

Shah

et al. 2024

AIMSPH

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<abstract> <p>Measles, a highly contagious viral disease, spreads primarily through respiratory droplets and can result in severe complications, often proving fatal, especially in children. In this article, we propose an algorithm to solve a system of fractional nonlinear equations that model the measles disease. We employ a fractional approach by using the Caputo operator and validate the model's by applying the Schauder and Banach fixed-point theory. The fractional derivatives, which constitute an essential part of the model can be treated precisely by using the Broyden and Haar wavelet collocation methods (HWCM). Furthermore, we evaluate the system's stability by implementing the Ulam-Hyers approach. The model takes into account multiple factors that influence virus transmission, and the HWCM offers an effective and precise solution for understanding insights into transmission dynamics through the use of fractional derivatives. We present the graphical results, which offer a comprehensive and invaluable perspective on how various parameters and fractional orders influence the behaviours of these compartments within the model. The study emphasizes the importance of modern techniques in understanding measles outbreaks, suggesting the methodology's applicability to various mathematical models. Simulations conducted by using MATLAB R2022a software demonstrate practical implementation, with the potential for extension to higher degrees with minor modifications. The simulation's findings clearly show the efficiency of the proposed approach and its application to further extend the field of mathematical modelling for infectious illnesses.</p> </abstract>

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“…Galerkin technique [40], Bernstein method [41], finite difference schemes method [42], Monotone iterative technique [43], wavelet method [44], and many others [39,45]. These methods vary in their level of complexity, but most of them are analytical methods, with only a few of them being numerical methods.…”

Section: Introductionmentioning

confidence: 99%

Employing a Fractional Basis Set to Solve Nonlinear Multidimensional Fractional Differential Equations

Rahman,

Bhatti,

Dimakis

2023

Mathematics

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Fractional-order partial differential equations have gained significant attention due to their wide range of applications in various fields. This paper employed a novel technique for solving nonlinear multidimensional fractional differential equations by means of a modified version of the Bernstein polynomials called the Bhatti-fractional polynomials basis set. The method involved approximating the desired solution and treated the resulting equation as a matrix equation. All fractional derivatives are considered in the Caputo sense. The resulting operational matrix was inverted, and the desired solution was obtained. The effectiveness of the method was demonstrated by solving two specific types of nonlinear multidimensional fractional differential equations. The results showed higher accuracy, with absolute errors ranging from 10−12 to 10−6 when compared with exact solutions. The proposed technique offered computational efficiency that could be implemented in various programming languages. The examples of two partial fractional differential equations were solved using Mathematica symbolic programming language, and the method showed potential for efficient resolution of fractional differential equations.

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

Cited by 15 publications

References 40 publications

Applications of Modified Bessel Polynomials to Solve a Nonlinear Chaotic Fractional-Order System in the Financial Market: Domain-Splitting Collocation Techniques

Applications of Modified Bessel Polynomials to Solve a Nonlinear Chaotic Fractional-Order System in the Financial Market: Domain-Splitting Collocation Techniques

On mathematical modelling of measles disease via collocation approach

Employing a Fractional Basis Set to Solve Nonlinear Multidimensional Fractional Differential Equations

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