An Effective Numerical Technique Based on the Tau Method for the Eigenvalue Problems

Attary, Maryam; Agarwal, Praveen

doi:10.1007/978-981-15-0430-3_12

Cited by 2 publications

(2 citation statements)

References 3 publications

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“…Note that for our purpose here, it is sufficient to discritize the equation using a finite difference method of uniform spacing. For other more sophisticated methods on solving the eigenvalue problems, we refer the interested readers to Attary and Agarwal 19 and the references therein.…”

Section: Numerical Examplesmentioning

confidence: 99%

On non‐local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

Chen

Kuo

Wang

et al. 2022

Math Methods in App Sciences

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A non‐local elliptic equation of Kirchhoff‐type −()a∫normalΩfalse|∇ufalse|2dx+1normalΔu=λffalse(xfalse)u+gfalse(xfalse)false|ufalse|γ−2u0.1em0.1emin0.5em0.1emnormalΩ$$ -\left(a{\int}_{\Omega}{\left|\nabla u\right|}^2 dx+1\right)\Delta u=\lambda f(x)u+g(x){\left|u\right|}^{\gamma -2}u\kern0.20em \mathrm{in}\kern0.60em \Omega $$ for a,λ>0$$ a,\lambda >0 $$ with Dirichlet boundary conditions is investigated for the cases where 1<γ<2$$ 1<\gamma <2 $$. It is well known that with the non‐local effect removed and f≡1$$ f\equiv 1 $$, a branch of positive solutions bifurcates from infinity at λ=λ1$$ \lambda ={\lambda}_1 $$ and no positive solution exists whenever λ>trueλ‾$$ \lambda >\overline{\lambda} $$ for some trueλ‾≥λ1$$ \overline{\lambda}\ge {\lambda}_1 $$ (see K. J. Brown, Calc. Var. 22, 483‐494, 2005),where λ1$$ {\lambda}_1 $$ is the principal eigenvalue of the linear problem −normalΔu=λu$$ -\Delta u=\lambda u $$. As a consequence of the non‐local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for λ>0$$ \lambda >0 $$. Moreover, regions with three positive solutions are found for small value of a$$ a $$. Comparisons are also made of the results here with those of the elliptic problem in the absence of the non‐local term under the same prescribed conditions using numerical simulations.

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Section: Numerical Examplesmentioning

confidence: 99%

On non‐local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

Chen

Kuo

Wang

et al. 2022

Math Methods in App Sciences

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“…Thus, the author [17] constructed quadrature generalized Gauss-Laguerre formulas for the boundary value problem with a fractional degree of an elliptic operator. Works [18,19] also propose approximative apparatuses for solving problems based on the Lagrangian polynomial interpolation [19]. For the control problem, the authors [20] present a Galerkin spectral scheme using weighted Jacobi polynomials and give optimal estimates of the spectral method error, and in [21] the finite element approximation of the fractional optimal time control problem with integral state constraint is investigated.…”

Section: Introductionmentioning

confidence: 99%

Calculation of fractional integrals using partial sums of Fourier series for structural mechanics problems

Galimyanov

Gorskaya

2021

E3S Web Conf.

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The goal of this study is to develop and apply an approximate method for calculating integrals that are part of models using Riemann-Liouville integrals, and to create a software product that allows such calculations for given functions. The main results of the study consist in the construction of a quadrature formula for an integral, and the cases where the density of the integral is a function from the spaces of continuous functions with generalized derivatives with weight and the Helder classes of functions with weight were considered. For the proposed quadrature formula we further investigated the error of its approximation in the spaces of continuous functions and quadratic-summing functions with weight. As a result of the study, effective error estimates of the approximating apparatus in the proposed classes of functions have been established. In addition, the approximated method has been implemented on the computer in the form of a program in the C language. The significance of the obtained results for the construction industry consists in the fact that when solving problems, including problems on finding the shapes of structures, taking into account the properties of materials, environmental changes, in the models of which the Riemann-Liouville integrals are used, it will be possible to apply an approximate approach, the quadrature formula proposed in the article.

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An Effective Numerical Technique Based on the Tau Method for the Eigenvalue Problems

Cited by 2 publications

References 3 publications

On non‐local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

On non‐local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

Calculation of fractional integrals using partial sums of Fourier series for structural mechanics problems

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