A computational method to find dual solutions of the one-dimensional Bratu problem

Karamollahi, Nasibeh; Loghmani, Ghasem Barid; Heydari, Mohammad

doi:10.1016/j.cam.2020.113309

Cited by 6 publications

(3 citation statements)

References 32 publications

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“…Хоёр шийдтэй үед дээд шийдийг тооцоолоход стандарт аргууд тохиромжгүй бөгөөд өндөр эрэмбийн нийлэлттэй аргууд ашиглах шаардлага тулгардаг. Сүүлийн жилүүдэд Брату бодлогын шийдийг судлах, тооцоолох ажлууд нэлээдгүй хийгдсэн [1][2][3][4][5][6][7][8][9][10][11]. Үүнээс [1] ажилд Брату бодлогын шийдийг төгсгөлөг ялгаврын арга, Коллокацийн арга, Адамын олон гишүүнтэд задлах арга, Онилох арга зэрэг аргуудаар тооцоолж, үр дүнг харьцуулан судалжээ.…”

Section: оршилunclassified

Artificial Neural Network Method for Solving of Bratu’s Problem

Badarch

Batgerel

2023

J. Inst. Math. Digit. Technol.

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The Bratu’s problem is widely used to model phenomenas such as heat transfer and combustion theory. For certain values of the parameters, there are 2 different solutions, and finding the lower solution is not difficult, and it is quite possible to apply standard mathematical methods for it calculation. However, finding the upper solution is difficult and requires the use of high-order convergent algorithms. In this study, the method of calculating the numerical solution of the Bratu’s problem using artificial neural networks is considered. When constructing the neural network, sinusoids were used as the activation function, and RMSprop (Root Mean Squared Propagation) was used as the optimization method. By doing so, its possible to calculate two solutions of the Bratu’s problem. Хиймэл Нейроны Сүлжээг Ашиглан Нэг Хэмжээст Брату Бодлогын Шийдийг Тооцоолох нь Хураангуй: Брату бодлогыг дулаан дамжуулалт, шаталтын процесс гэх мэт үзэгдлүүдийг загварчлахад өргөн ашигладаг. Тэгшитгэл параметрийн тодорхой утгуудад хоёр шийдтэй бөгөөд доод шийдийг олох нь төвөггүй, тооцон бодох математикийн стандарт аргуудыг хэрэглэх бүрэн боломжтой. Харин дээд шийдийг олох нь бэрхшээлтэй бөгөөд өндөр эрэмбийн нийлэлттэй алгоритмуудыг ашиглах шаардлага тулгардаг. Энэхүү судалгаанд Брату бодлогын тоон ший- дийг хиймэл нейроны сүлжээ ашиглан тооцоолох аргыг авч үзлээ. Нейроны сүлжээг байгуулахдаа идэвхжилтийн функцээр синусоидийг, оновчлолын аргаар RMSprop (Root Mean Squared Propagation) аргыг ашиглав. Ингэснээр Брату бодлогын хоёр шийдийг тооцоолох боломжтой болов. Түлхүүр үгс: Машин сургалт, Дифференциал тэгшитгэл, Захын нөхцөлт бодлого, Стандарт бус төгсгөлөг ялгаврын арга

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Section: оршилunclassified

Artificial Neural Network Method for Solving of Bratu’s Problem

Badarch

Batgerel

2023

J. Inst. Math. Digit. Technol.

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“…Tomar et al [22] introduced an efficient analytical iterative method that obtains a semi-analytical solution for the Bratu-type problem. Karamollahi et al [23] used Hermite interpolation to approximate dual solutions for the strongly nonlinear one-dimensional Bratu problem. Jator et al [24] introduced a new numerical scheme to solve Bratu's problem.…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System

Ene,

Pop

2024

Mathematics

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Mathematical models and numerical simulations are necessary to understand the functions of biological rhythms, to comprehend the transition from simple to complex behavior and to delineate the conditions under which they arise. The aim of this work is to investigate the Ro¨ssler-type system. This system could be proposed as a theoretical model for biological rhythms, generalizing this formula for chaotic behavior. It is assumed that the Ro¨ssler-type system has a Hamilton–Poisson realization. To semi-analytically solve this system, a Bratu-type equation was explored. The approximate closed-form solutions are obtained using the Optimal Parametric Iteration Method (OPIM) using only one iteration. The advantages of this analytical procedure are reflected through a comparison between the analytical and corresponding numerical results. The obtained results are in a good agreement with the numerical results, and they highlight that our procedure is effective, accurate and usefully for implementation in applicationssuch as an oscillator with cubic and harmonic restoring forces, the Thomas–Fermi equation and the Lotka–Voltera model with three species.

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“…For λ > 0, the exact solution of problem (1)–(2) is given by Karamollahi et al (2021), Liu et al (2013), Ragb et al (2017), Swaminathan et al (2021): …”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

Umesh

2023

HFF

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Purpose This paper aims to discuss a new form of the Adomian decomposition technique for the numerical treatment of Bratu’s type one-dimensional boundary value problems (BVPs). Moreover, the author also addresses convergence and error analysis for the completeness of the proposed technique. Design/methodology/approach First, the author discusses the standard Adomian decomposition method and an algorithm based on Duan’s corollary and Rach’s rule for the fast calculation of the Adomian polynomials. Then, a new form of the Adomian decomposition technique is present for the numerical simulation of Bratu’s BVPs. Findings The reliability and validity of the proposed technique are examined by calculating the absolute errors of Bratu’s problem for some different values of Bratu parameter λ. Numerical simulation demonstrates that the proposed technique yields higher accuracy than the Bessel collocation and other known methods. Originality/value Unlike the other methods, the proposed technique does not need linearization, discretization or perturbation to handle the non-linear problems. So, the results obtained by the present technique are more physically realistic.

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A computational method to find dual solutions of the one-dimensional Bratu problem

Cited by 6 publications

References 32 publications

Artificial Neural Network Method for Solving of Bratu’s Problem

Artificial Neural Network Method for Solving of Bratu’s Problem

Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

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