On Kirchhoff's Model of Parabolic Type

Kundu, Sudeep; Pani, Amiya K.; Khebchareon, Morrakot

doi:10.1080/01630563.2016.1176930

Cited by 15 publications

(5 citation statements)

References 4 publications

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“…To overcome this problem, author in his work proposed a modified Newton-Raphson method for stationary version of Equation 4and obtained optimal rate of convergence in H 1 norm. Further, Kundu et al 22 considered a particular case of diffusion coefficient in Equation 4…”

Section: Introductionmentioning

confidence: 99%

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Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

Kumar

Sista

Sreenadh

2020

Math Methods in App Sciences

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The aim of this paper is to study parabolic integro‐differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.

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Section: Introductionmentioning

confidence: 99%

“…norm by employing linearized backward Euler-Galerkin method. The main objective of this article is to analyze a more general case of the problem studied in Kundu et al 22 by taking diffusion coefficient M (…”

Section: Introductionmentioning

confidence: 99%

Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

Kumar

Sista

Sreenadh

2020

Math Methods in App Sciences

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“…The function u is a descriptive population density (e.g., bacteria) spread. According to article [26], we find that model ( 1) is a type of Kirchhoff equation, arising in vibration theory; see, for example, [27].…”

Section: Introductionmentioning

confidence: 99%

Final Value Problem for Parabolic Equation with Fractional Laplacian and Kirchhoff’s Term

Luc¹,

Kumar²,

Long³

et al. 2021

Journal of Function Spaces

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In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero when δ tends to zero.

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“…For related numerical methods, a good number of article is devoted to strongly nonlinear elliptic problems, see, [4], [10], [14], [16], [17], [25], [28], [29], [35] and references, there in. However, there seems to be less number of papers available in literature on numerical approximation to strongly nonlinear parabolic problems, see, [3], [19], [30] and [22], etc. The more relevant article is [3], where conforming FEM is applied to the problem (1.1)-(1.3) and optimal error estimates in L ∞ (L 2 ) are derived using piecewise polynomial of degree r ≥ 2 for d = 2 and for d = 3, r ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Negative norm estimates and superconvergence results in Galerkin method for strongly nonlinear parabolic problems

Pany¹,

Khebchareon²,

Pani³

2021

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The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree r ≥ 1 are used, which improve upon earlier results of Axelsson [Numer. Math. 28 (1977), pp. 1-14] requiring for 2d r ≥ 2 and for 3d r ≥ 3. Based on quasi-projection technique introduced by Douglas et al. [Math. Comp.32 (1978),pp. 345-362], superconvergence result for the error between Galerkin approximation and approximation through quasi-projection is established for the semidiscrete Galerkin scheme. Further, a priori error estimates in Sobolev spaces of negative index are derived. Moreover, in a single space variable, nodal superconvergence results between the true solution and Galerkin approximation are established.

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On Kirchhoff's Model of Parabolic Type

Cited by 15 publications

References 4 publications

Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

Final Value Problem for Parabolic Equation with Fractional Laplacian and Kirchhoff’s Term

Negative norm estimates and superconvergence results in Galerkin method for strongly nonlinear parabolic problems

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