New formulation of the orthonormal Bernoulli polynomials for solving the variable-order time fractional coupled Boussinesq–Burger’s equations

Heydari, Mohammad Hossein; Avazzadeh, Z.

doi:10.1007/s00366-020-01007-w

Cited by 18 publications

(5 citation statements)

References 19 publications

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“…Various approaches were applied to explain the space-time fractional order coupled type BB equation for example M. H. Heydari and Z. Avazzadeh were solving this equation using the Orthonormal Bernoulli polynomials (OBPs) with derivative in the Caputo form [ 31 ], Kumar et. al.…”

Section: Commencement and Forewordmentioning

confidence: 99%

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Zaman¹,

Arefin²,

Akbar

et al. 2023

PLoS ONE

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Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq–Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves.

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Section: Commencement and Forewordmentioning

confidence: 99%

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Zaman¹,

Arefin²,

Akbar

et al. 2023

PLoS ONE

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“…A set of ( M + 1) orthonormal Bernoulli polynomials is generated on [0, b ] by the following formula (Heydari and Avazzadeh, 2020)where…”

Section: Orthonormal Piecewise Bernoulli Functionsmentioning

confidence: 99%

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

Heydari

Razzaghi

Avazzadeh

2022

Journal of Vibration and Control

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In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

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“…13 Since it is often impossible to solve VO fractional problems, analytically, varied numerical techniques have been used for such problems. Some numerical procedures for solving these class of problems are including the orthonormal Bernoulli polynomials method, 14 the Chebyshev collocation method, 15 the sixth-kind Chebyshev polynomials technique, 16 the meshless approach, 17 the Lagrange polynomials method, 18 the transcendental Bernstein expansion method, 19 and the Lagrangian piecewise interpolation technique. 20 The regularized long wave Burgers (RLW-Burgers) equation as a well-known nonlinear partial differential equation plays an important role in describing surface water waves propagation in a channel.…”

Section: Introductionmentioning

confidence: 99%

Third‐kind Chebyshev cardinal functions for variable‐order time fractional RLW‐Burgers equation

Heydari

Razzaghi

2022

Math Methods in App Sciences

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In this article, the Caputo type variable-order fractional differentiation is utilized for defining a novel fractional form of the regularized long wave Burgers problem which arises in several physical applications. The third-kind Chebyshev cardinal functions are generated and some matrix relationships related to them are derived. A numerical procedure dependent on these polynomial cardinal functions is developed for the introduced equation. This scheme converts the primary problem solving into finding an algebraic system solution with low computations using approximating the solution of the equation by the expressed cardinal functions. The accuracy of the presented procedure is investigated in some examples. The presented scheme is straightforward and powerful and can be adopted for many other nonlinear fractional problems.

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New formulation of the orthonormal Bernoulli polynomials for solving the variable-order time fractional coupled Boussinesq–Burger’s equations

Cited by 18 publications

References 19 publications

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

Third‐kind Chebyshev cardinal functions for variable‐order time fractional RLW‐Burgers equation

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