Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

Jani, Mostafa; Babolian, E.; Javadi, Shahnam

doi:10.1002/mma.4551

Cited by 11 publications

(9 citation statements)

References 30 publications

(37 reference statements)

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“…Remark For Problems ()–() with the first Dirichlet boundary conditions (), the following basis for the test space in which

{\overset{ψ}{true}}_{i} false(x false) {false|}_{x = 0,1} = \frac{d}{d x} {\overset{ψ}{true}}_{i} false(x false) {false|}_{x = 0,1} = 0

may be used 32 :

{\tilde{ψ}}_{i} = ψ_{i} + \frac{i + 3}{N - i + 2} (4 ψ_{i + 1} + \frac{i + 4}{N - i + 1} (() 6 ψ_{i + 2} + \frac{i + 5}{N - i} (4 ψ_{i + 3} + \frac{i + 6}{N - i - 1} ψ_{i + 4}))), 0 \leq i \leq N - 4 .

Now we are seeking an approximate solution of () as

u_{N}^{k + 1} false(x false) = true \sum_{j = 2}^{N - 2} c_{j}^{k + 1} {\overset{ϕ}{true}}_{j} false(x false),

where the coefficients are obtained by taking the test functions as

v_{N} = {\overset{ψ}{true}}_{i}

, introduced in Theorem 3, in ()

true \sum_{j = 2}^{N - 2} c_{j}^{k + 1} false(false(ω_{0} + μ false)

…”

Section: Bernstein Petrov–galerkin Approachmentioning

confidence: 99%

A Petrov–Galerkin spectral method for the numerical simulation and analysis of fractional anomalous diffusion

Jani

Babolian

Bhatta

2020

Math Methods in App Sciences

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Anomalous diffusion problems are used to describe the evolution of particle's motion in crowded environments with many applications, such as modeling the intracellular transport and disordered media. In the present paper, we develop a Petrov-Galerkin spectral method for the fourth-order anomalous fractional diffusion equations. For the dimension reduction, we use a discretization in time by the convolution quadrature. We then introduce the basis sets for the trial-and-test spaces using modal Bernstein basis functions with a presentation of the method in a weak spectral formulation along with a discussion of the structure of the resulting systems, the convergence, and stability of the proposed method. The theoretical results are supported by illustrating some numerical examples.

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“…Remark For Problems ()–() with the first Dirichlet boundary conditions (), the following basis for the test space in which

{\overset{ψ}{true}}_{i} false(x false) {false|}_{x = 0,1} = \frac{d}{d x} {\overset{ψ}{true}}_{i} false(x false) {false|}_{x = 0,1} = 0

may be used 32 :

{\tilde{ψ}}_{i} = ψ_{i} + \frac{i + 3}{N - i + 2} (4 ψ_{i + 1} + \frac{i + 4}{N - i + 1} (() 6 ψ_{i + 2} + \frac{i + 5}{N - i} (4 ψ_{i + 3} + \frac{i + 6}{N - i - 1} ψ_{i + 4}))), 0 \leq i \leq N - 4 .

Now we are seeking an approximate solution of () as

u_{N}^{k + 1} false(x false) = true \sum_{j = 2}^{N - 2} c_{j}^{k + 1} {\overset{ϕ}{true}}_{j} false(x false),

where the coefficients are obtained by taking the test functions as

v_{N} = {\overset{ψ}{true}}_{i}

, introduced in Theorem 3, in ()

true \sum_{j = 2}^{N - 2} c_{j}^{k + 1} false(false(ω_{0} + μ false)

…”

Section: Bernstein Petrov–galerkin Approachmentioning

confidence: 99%

A Petrov–Galerkin spectral method for the numerical simulation and analysis of fractional anomalous diffusion

Jani

Babolian

Bhatta

2020

Math Methods in App Sciences

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“…21 деформівного твердого тіла. Ефективність використання аналітико-числових методів показана при розв'язуванні багатьох задач сучасної математичної фізики: [2] -використовують інтерполяційні функції Лобатто -Лежандра в задачі геодезичної гравіметрії; [3] -використовують усічені ряди Фур'є в розв'язку задачі поширення теплового потоку в спіральних трубах; [4] -при вивченні руху електронів в атомах та іонах у сферичній системі координат в якості базису були використані гамільтонові власні функції і сферичні гармоніки, тобто пов'язані з ними функції Лежандра; [5] -при розв'язуванні рівняння адвекціїдисперсії та дифузії з дробовою похідною за часом використовували дуальний базис Бернштейна.…”

Section: проектирование летательных аппаратовunclassified

Числове Дослідження Базисних Систем При Розв’язуванні Аналітико-Числовими Методами Крайових Задач На Відрізку

Халілов¹,

Ткаченко²,

Бондарева³

et al. 2019

aktt

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Spectral methods have indisputable advantages over numerical methods in solving various problems of mathematical physics. The advantages are the high convergence and accuracy of approximate solutions, which is most relevant for calculating the strength of aerospace machinery. The problem of choosing basic functions inevitably arises when applying spectral methods. The problem is that, in addition to providing exponential convergence, the system of basic functions has to satisfy some other requirements: stability of approximate solutions and procedures for their obtaining, reduction of calculations, convenience, and some more. This paper compares eleven basic systems: the systems constructed in the form of linear combinations of Legendre polynomials that satisfy either only the main boundary conditions or the main and natural ones, similar systems constructed using Chebyshev polynomials, and the functions proposed by Khalilov S. A., systems of Lagrange – Lobatto interpolation polynomials using the Legendre and Chebyshev interpolation points, system of trigonometric functions, exponentiation, and system of finite functions of the finite element method. The convergence speed of the approximate solution to the exact one, the error in the equations of boundary value problems, and the condition numbers of matrices of linear algebraic equation systems, which arise when using variational, projection and collocation methods, were compared. The study performed on three test problems modeling beam bending: classic beam bending under unevenly distributed load, bending of additionally stretched beam on the elastic basis and geometrically nonlinear bending. The impact of the Gibbs effect on the approximate solution convergence is investigated. Among the considered basic systems, the system of basis functions in the form of linear combinations of Legendre polynomials has proved to be the best, as they satisfy all boundary conditions. This basis leads to the highest speed at which approximate solution approaches the exact one, the error in the equations approaches to zero, and also it has the smallest increase in the condition number with the increase in the order of SLE matrices, which appear due to variational and projection methods. The finite functions of the finite element method have proved to be the worst in terms of accuracy and convergence.

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“…The Bernstein polynomials have also been modified the Adomian decomposition [31] and Laplace decomposition method [33] for solving linear and nonlinear differential equations. The spectral Petrov-Galerkin method [20] has been applied for numerical solution of fractional partial differential equations that are one of the most popular and remarkable equations in the science world. Discretization method [13] that is other numerical method has been applied to the Volterra-Fredholm integral equations by utilizing the Bernstein polynomials.…”

Section: Introductionmentioning

confidence: 99%

Bernstein Operator Approach for Solving Linear Differential Equations

Acar

Daşçıoğlu²

2021

Mathematical Sciences and Applications E-Notes

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Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

Cited by 11 publications

References 30 publications

A Petrov–Galerkin spectral method for the numerical simulation and analysis of fractional anomalous diffusion

A Petrov–Galerkin spectral method for the numerical simulation and analysis of fractional anomalous diffusion

Числове Дослідження Базисних Систем При Розв’язуванні Аналітико-Числовими Методами Крайових Задач На Відрізку

Bernstein Operator Approach for Solving Linear Differential Equations

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