Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Singh, Harendra; Pandey, Rajesh K.; Srivástava, H. M.

doi:10.3390/math7030224

Cited by 41 publications

(13 citation statements)

References 38 publications

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“…Here, in this section, we present an approach for solving the Sturm-Liouville problem arising from a stability analysis for traveling waves. The idea behind this approach is inspired by some results for the Legendre functions and the Legendre polynomials (see also the recent investigations [23,24] involving applications of the substantially more general Jacobi polynomials).…”

Section: A Direct Approach For Analyzing the Stability Of Traveling Wmentioning

confidence: 99%

Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials

2020

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One of the tools and techniques concerned with the stability of nonlinear waves is the Evans function which is an analytic function whose zeros give the eigenvalues of the linearized operator. Here, in this paper, we propose a direct approach, which is based essentially upon constructing the eigenfunction solution of the perturbed equation based upon the topological invariance in conjunction with usage of the Legendre polynomials, which have presumably not considered in the literature thus far. The associated Legendre eigenvalue problem arising from the stability analysis of traveling waves solutions is systematically studied here. The present work is of considerable interest in the engineering sciences as well as the mathematical and physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product. This can be controlled by depicting the initial concentration of the reactant, which is determined by its value at the bifurcation point. This analysis leads to the point separating stable and unstable solutions. As far as chemical reactions are described by reaction-diffusion equations, this specific concentration can be found mathematically. On the other hand, the study of stability analysis of solutions may depict whether or not a soliton pulse is well-propagated in fiber optics. This can, and should, be carried out by finding the solutions of the coupled nonlinear Schrödinger equations and by analyzing the stability of these solutions.

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Section: A Direct Approach For Analyzing the Stability Of Traveling Wmentioning

confidence: 99%

Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials

2020

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“…Modern calculus has generalized the integer order calculus of differentiation and integration to rational or complex numbers, describing the situation between two integer values as in [49] , [50] , [51] . Up to now, various real-world problems have been modeled by integer-order differential equations, like population model, logistic population model, HIV, SEIR, TB, Cancer model, Predator–prey model, etc.…”

Section: Introductionmentioning

confidence: 99%

Investigation of fractal-fractional order model of COVID-19 in Pakistan under Atangana-Baleanu Caputo (ABC) derivative

et al. 2021

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This manuscript addressing the dynamics of fractal-fractional type modified SEIR model under Atangana-Baleanu Caputo (ABC) derivative of fractional order and fractal dimension for the available data in Pakistan. The proposed model has been investigated for qualitative analysis by applying the theory of non-linear functional analysis along with fixed point theory. The fractional Adams–bashforth iterative techniques have been applied for the numerical solution of the said model. The Ulam-Hyers (UH) stability techniques have been derived for the stability of the considered model. The simulation of all compartments has been drawn against the available data of covid-19 in Pakistan. The whole study of this manuscript illustrates that control of the effective transmission rate is necessary for stoping the transmission of the outbreak. This means that everyone in the society must change their behavior towards self-protection by keeping most of the precautionary measures sufficient for controlling covid-19.

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“…Non-integer order calculus has provided the information of the spectrum at a fractional or rational value between the integer orders [19] , [20] . The representation of different real phenomena have formulated by fractional order integral or differential equation like mathematical fractional order model for microorganism population, a logistic non-linear model for the human population, tuberculosis model, dingy problem, hepatitis B, C models and the basic Lotka-Volterra models being the basis of all infectious problems [21] , [22] , [23] , [24] , [25] , [26] .…”

Section: Introductionmentioning

confidence: 99%

Fractal-fractional order mathematical vaccine model of COVID-19 under non-singular kernel

Algehyne

Ibrahim

2021

Chaos, Solitons & Fractals

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In this paper,the severe acute respiratory syndrome coronavirus (SARS-CoV-2) or COVID-19 is researched by employing mathematical analysis under modern calculus. In this context, the dynamical behavior of an arbitrary order p and fractal dimensional q problem of COVID-19 under Atangana Bleanu Capute (ABC) operator for the three cities, namely, Santos, Campinas, and Sao Paulo of Brazil are investigated as a case-study. The considered problem is analyzed for at least one solution and unique solution by the applications of the theorems of fixed point and non-linear functional analysis. The Ulam-Hyres stability condition via nonlinear functional analysis for the given system is derived. In order to perform the numerical simulation, a two-step fractional type, Lagrange plynomial (Adams Bashforth technique) is utilized for the present system. MATLAB simulation tools have been used for testing different fractal fractional orders considering the data of aforementioned three regions. The analysis of the results finally infer that, for all these three regions, the smaller order values provide better constraints than the larger order values.

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Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Cited by 41 publications

References 38 publications

Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials

Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials

Investigation of fractal-fractional order model of COVID-19 in Pakistan under Atangana-Baleanu Caputo (ABC) derivative

Fractal-fractional order mathematical vaccine model of COVID-19 under non-singular kernel

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