Bernstein polynomials method and it’s error analysis for solving nonlinear problems in the calculus of variations: Convergence analysis via residual function

Bataineh, A. Sami

doi:10.2298/fil1804379b

Cited by 8 publications

(6 citation statements)

References 19 publications

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“…If problem (14) has a unique C 2 [0, 1] continuous solutionū, then the approximate solution obtained by the control-point-based method converges to the exact solutionū as the degree of the approximate solution tends to infinity.…”

Section: Theoremmentioning

confidence: 99%

“…This solution displays a bifurcation pattern, which only characterizes nonlinear differential equations. In fact, the following one-dimensional Bratu's problem is utilized in various types of applications such as the fuel ignition model of the thermal combustion theory, radioactive heat transfer and nanotechnology (see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). In this paper, we have the following Bratu's boundary value problem:…”

Section: Introductionmentioning

confidence: 99%

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A new approach for solving Bratu’s problem

Ghomanjani

Shateyi

2019

Demonstratio Mathematica

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A numerical technique for one-dimensional Bratu’s problem is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are exhibited to verify the efficiency and accuracy of the proposed technique. In this sequel, the obtained error was shown between the proposed technique, Chebyshev wavelets, and Legendre wavelets. The results display that this technique is accurate.

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Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new approach for solving Bratu’s problem

Ghomanjani

Shateyi

2019

Demonstratio Mathematica

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“…We can find the function approximation based on Bernstein polynomials. A function f (t), f (t) ∈ L 2 [0, 1] can be expressed in terms of orthonormal Bernstein polynomials [47]:…”

Section: Bernstein Polynomials and Function Approximationmentioning

confidence: 99%

“…In order to show the effectiveness of our method, a residual function of a linear time varying system in the Banach space for the values of k is adopted to interpret the convergence of the Bernstein polynomials solution as described in [47,53]. Suppose f k (t) are the approximate solution of (10), we write the residual function as…”

Section: Convergence Analysismentioning

confidence: 99%

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Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations

2019

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This paper discusses the applications of numerical inversion of the Laplace transform method based on the Bernstein operational matrix to find the solution to a class of fractional differential equations. By the use of Laplace transform, fractional differential equations are firstly converted to system of algebraic equations then the numerical inverse of a Laplace transform is adopted to find the unknown function in the equation by expanding it in a Bernstein series. The advantages and computational implications of the proposed technique are discussed and verified in some numerical examples by comparing the results with some existing methods. We have also combined our technique to the standard Laplace Adomian decomposition method for solving nonlinear fractional order differential equations. The method is given with error estimation and convergence criterion that exclude the validity of our method.

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A hybrid collocation method for solving highly nonlinear boundary value problems

Adewumi

Akindeinde

Aderogba

et al. 2020

Heliyon

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In this article, a hybrid collocation method for solving highly nonlinear boundary value problems is presented. This hybrid method combines Chebyshev collocation method with Laplace and differential transform methods to obtain approximate solutions of some highly nonlinear two-point boundary value problems of ordinary differential equations. The efficiency of the method is demonstrated by applying it to ordinary differential equations modelling Darcy-Brinkman-Forchheimer momentum problem, laminar viscous flow problem in a semi-porous channel subject to transverse magnetic field, fin problem with a temperature-dependent thermal conductivity, transformed equations modelling two-dimensional viscous flow problem in a rectangular domain bounded by two moving porous walls and two-dimensional constant speed squeezing flow of a viscous fluid between two approaching parallel plates. The results obtained are compared with the existing methods and the results show that the new method is quite reasonable, accurate and efficient.

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Bernstein polynomials method and it’s error analysis for solving nonlinear problems in the calculus of variations: Convergence analysis via residual function

Cited by 8 publications

References 19 publications

A new approach for solving Bratu’s problem

A new approach for solving Bratu’s problem

Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations

A hybrid collocation method for solving highly nonlinear boundary value problems

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