A numerical scheme for a class of generalized Burgers' equation based on Haar wavelet nonstandard finite difference method

Verma, Amrisha; Rawani, Mukesh Kumar; Cattani, Carlo

doi:10.1016/j.apnum.2021.05.019

Cited by 16 publications

(4 citation statements)

References 53 publications

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“…Therefore, they are efficient and appropriate to simulate behaviour of dynamical differential equation models over long time periods. Nowadays, NSFD schemes have become an efficient approach for numerically solving real-world problems (see, for example, [1,2,4,5,10,13,14,15,31,32,47,48,57,58]). Recently, we have developed the Mickens' methodology to construct NSFD schemes for mathematical models of phenomena and processes coming from sciences and technology like biology, ecology, or other natural sciences [16,17,18,19,20,21,25,26,27,28,29].…”

Section: Introductionmentioning

confidence: 99%

A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

Hoang

2023

Preprint

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Although Coronavirus disease 2019 (COVID-19) pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has been controlled and prevented, mathematical modeling and analysis of transmission dynamics of COVID-19 still plays an essential role not only in the post COVID-19 era but also in the study of infectious diseases. This is an important foundation to propose effective strategies and measures for controlling diseases and projecting public health. This work is devoted to proposing and analyzing a new mathematical study for transmission dynamics of COVID-19. We first introduce a generalized SEIR epidemic model that use general nonlinear incidence rates to describe the “psychological” effect. After that, a rigorous mathematical analysis for the proposed COVID-19 model is performed. We establish the positivity and boundedness, calculate the basic reproduction number, determine possible (disease-free and endemic-disease) equilibrium points and investigate their asymptotic stability properties of the SEIR model. The obtained results improve and extend an SEIR model constructed in a recent work. For the purpose of numerical simulation, the Mickens’ methodology is applied to construct a dynamically consistent nonstandard finite difference (NSFD) model for the proposed SEIR epidemic model. The constructed NSFD scheme has the ability to provide reliable approximations that not only preserve the dynamical properties of the SEIR model for all the values of the step size but also are easy to be implemented. Finally, a set of illustrative numerical experiments is conducted to support the theoretical findings and to confirm advantages of the NSFD scheme over some well-known standard ones.

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Section: Introductionmentioning

confidence: 99%

A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

Hoang

2023

Preprint

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“…On the other hand, NSFD schemes are simple, effective and can be applied to solve a broad class of differential equations. For these reasons, NSFD schemes have been widely used in solving differential equation models arising in real-world applications nowadays [1,2,8,9,10,20,21,22,30,33,34,35,66,75,77,78]. In [23,24,25,26,43,44,45,46,47,48],…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis and numerical simulation of a new fractional-order two-stage species model with recruitment

Hoang

2023

Preprint

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The purpose of this work is to propose and analyze a new fractional-order two-stage species model with recruitment to discover memory effects on population dynamics. This fractional-order model is constructed by incorporating a well-known integer-order two-stage species model with the Caputo fractional derivative. Firstly, the positivity and boundedness of solutions of the proposed model are investigated by using some standard comparison results for fractional differential equations. Next, a simple approach is utilized to study stability properties of the fractional-order model. This approach is based on the Lyapunov stability theory and Barbalat’s lemma in combination with some nonstandard techniques for fractional dynamical systems. More clearly, we use general quadratic Lyapunov candidate functions and combine them with characteristics of quadratic forms associated with real matrices to establish the stability properties. As an important consequence, global asymptotic stability, uniform and Mittag-Leffler stability and therefore, population dynamics of the proposed fractional-order model are analyzed rigorously. Finally, we extend the Mickens’ methodology to construct a dynamically consistent nonstandard finite difference (NSFD) scheme for the purpose of numerical simulation of the fractional-order model. It is proved that the NSFD scheme preserves the positivity and boundedness of the fractional-order model regardless of the values of the step size; moreover, it is also simple and efficient. The theoretical results and advantages of the NSFD scheme are supported by illustrative numerical experiments. The experiments provide strong evidence, which shows that the numerical results are consistent with the theoretical ones.

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“…Shukla and Kumar [54] used a combination of the uniform Haar wavelet analysis and the Crank-Nicolson finite difference approach to numerically solve the B-H problem. Recently, Verma et al [55] developed a numerical technique based on the uniform Haar wavelet and non-standard finite difference scheme for solving a class of extended Burgers equations. A higher-order non-uniform Haar wavelet approach was also suggested by Ratas et al [56] for solving nonlinear PDEs.…”

Section: Introductionmentioning

confidence: 99%

Two-Dimensional Uniform and Non-Uniform Haar Wavelet Collocation Approach for a Class of Nonlinear PDEs

Kumar,

Verma,

Agarwal

2023

Computation

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In this paper, we introduce a novel approach employing two-dimensional uniform and non-uniform Haar wavelet collocation methods to effectively solve the generalized Burgers–Huxley and Burgers–Fisher equations. The demonstrated method exhibits an impressive quartic convergence rate. Several test problems are presented to exemplify the accuracy and efficiency of this proposed approach. Our results exhibit exceptional accuracy even with a minimal number of spatial divisions. Additionally, we conduct a comparative analysis of our results with existing methods.

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A numerical scheme for a class of generalized Burgers' equation based on Haar wavelet nonstandard finite difference method

Cited by 16 publications

References 53 publications

A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

Dynamical analysis and numerical simulation of a new fractional-order two-stage species model with recruitment

Two-Dimensional Uniform and Non-Uniform Haar Wavelet Collocation Approach for a Class of Nonlinear PDEs

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