Multivalued behavior for a two-level system using Homotopy Analysis Method

Aquino, Angelo I.; Bo-ot, L.Ma.

doi:10.1016/j.physa.2015.09.057

Cited by 8 publications

(6 citation statements)

References 14 publications

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“…Whereas the series solution ∑ ∞ n=0 n (x, t) converges, where n (x, t) is governed by (15) under the definition (16), the limit of the series is an exact solution of Equation 1.…”

Section: The Linearization-based Approach Of Hammentioning

confidence: 99%

“…The actual success of HAM is greatly depends on the appropriate selection of these parameters. The theory and applications of the method has been developed recently to deal with numerous types of non-linear problems in applied science fields (see previous studies [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references therein). In previous studies, [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] the application of the method has been expanded to deal with non-linear fractional differential models.…”

Section: Introductionmentioning

confidence: 99%

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A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

Odibat

Băleanu

2019

Math Methods in App Sciences

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In this paper, a novel approach, namely, the linearization-based approach of homotopy analysis method, to analytically treat non-linear time-fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non-linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non-linear time-fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization-based approach reduces the computational complexity and improves the performance of the homotopy analysis method. KEYWORDS homotopy analysis method, linearization-based approach of HAM, series solution, time-fractional parabolic PDE MSC CLASSIFICATION 26A33; 34E10; 35K15; 41A58 INTRODUCTIONThe homotopy analysis method (HAM), introduced in previous studies, 1-5 is an approximation technique used to analytically solve various types of non-linear differential equations. This method utilizes homotopy in order to deform a continuous mapping with free parameters of an initial guessed approximation to the exact solution of the considered problem. The actual success of HAM is greatly depends on the appropriate selection of these parameters. The theory and applications of the method has been developed recently to deal with numerous types of non-linear problems in applied science fields (see previous studies 6-20 and references therein). In previous studies, 21-40 the application of the method has been expanded to deal with non-linear fractional differential models. These extensions, which generalize the HAM, were successfully implemented to get precise approximate solutions of non-linear time-and space-time fractional differential equations accurately and conveniently. Furthermore, some approaches which are designed optimally to improve the efficiency of the HAM have been introduced. 41-51 These approaches present techniques by which one may select best or appropriate parameters. As observed in other work, 41-51 the employment of the optimal parameters within HAM accelerate the rapid convergence of the HAM series solutions.The flexibility of the HAM in selecting the auxiliary control parameter, the initial guess, the auxiliary linear operator and the auxiliary function is one of the main merits of the method. Liao 1 mentioned that it is necessary to introduce mathematical results that propose new ideas for the appropriate selection of the auxiliary control parameter, the initial guess and the auxiliary linear operator. Therefore, the main objective of this paper is to implement Taylor series approximation of the non-linear equation in order to extract an optimized linear operator. Then, to construct a favourable homotopy Math Meth Appl Sci. 2019;42:7222-7232.

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“…Whereas the series solution ∑ ∞ n=0 n (x, t) converges, where n (x, t) is governed by (15) under the definition (16), the limit of the series is an exact solution of Equation 1.…”

Section: The Linearization-based Approach Of Hammentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

Odibat

Băleanu

2019

Math Methods in App Sciences

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“…The results obtained by homotopy analysis method (HAM) are preferred than the numerical solutions in perspective of the following points [31][32][33][34][35][36][37][38][39]: -The HAM gives the solutions within the domain of interest at each point while the numerical solutions hold just for discrete points in the domain. -Algebraically developed approximate solutions require less effort and having a sensible measure of precision when compared to numerical solution which are more convenient for the scientist, an engineer or an applied mathematician.…”

Section: Introductionmentioning

confidence: 99%

Non-linear thermal radiation and magnetic field effects on the flow Carreau nanofluid with convective conditions

et al. 2020

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Main features of the present analysis is to investigate the MHD non-linear mixed convection flow of Carreau nanofluid. Flow is due to stretching sheet with thermal and solutal convective conditions. Intention in present analysis is to develop a model for nanomaterial. The non-linear ordinary differential systems are obtained. Homotopy algorithm leads to solutions development. Velocity, temperature, nanoparticles concentration, surface drag force, and heat and mass transfer rate are displayed and argued. It is revealed that qualitative behaviors of velocity and layer thickness are reverse for material and magnetic parameters. Temperature field and heat transfer rate are similar observation for thermal Biot numbers. Moreover qualitative behaviors of nanoparticles concentration and mass transfer rate are reverse for larger Brownian motion.

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“…Hetmaniok et al [Hetmaniok, Slota, Witula et al (2015)] used it to solve the one-phase inverse Stefan problem. It was also used to explore the multi-valued behavior for a twolevel system by Aquino et al [Aquino and Boot (2016)]. Curato et al [Curato, Gatheral and Lillo (2016)] suggested a discrete homotopy analysis for solving the nonlinear transient market problem.…”

Section: Introductionmentioning

confidence: 99%

Equivalence of Ratio and Residual Approaches in the Homotopy Analysis Method and Some Applications in Nonlinear Science and Engineering

Türkyılmazoğlu¹

2019

Computer Modeling in Engineering &Amp; Sciences

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A ratio approach based on the simple ratio test associated with the terms of homotopy series was proposed by the author in the previous publications. It was shown in the latter through various comparative physical models that the ratio approach of identifying the range of the convergence control parameter and also an optimal value for it in the homotopy analysis method is a promising alternative to the classically used h-level curves or to the minimizing the residual (squared) error. A mathematical analysis is targeted here to prove the equivalence of both the ratio approach and the traditional residual approach, especially regarding the root-finding problems via the homotopy analysis method. Examples are provided to further justify this. Moreover, it is conjectured that every nonlinear differential equation can be considered as a root-finding problem by plugging a parameter in it from a physical viewpoint. Two examples from the boundary and initial and value problems are provided to verify this assertion. Hence, besides the advantages as deciphered in the previous publications, the feasibility of the ratio approach over the traditional residual approach is made clearer in this paper.

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Multivalued behavior for a two-level system using Homotopy Analysis Method

Cited by 8 publications

References 14 publications

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

Non-linear thermal radiation and magnetic field effects on the flow Carreau nanofluid with convective conditions

Equivalence of Ratio and Residual Approaches in the Homotopy Analysis Method and Some Applications in Nonlinear Science and Engineering

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