A boundary-only integral equation method for parabolic problems of another class of anisotropic functionally graded materials

Azis, Moh. Ivan; Abbaszadeh, Mostafa; Dehghan, Mehdi; Solekhudin, Imam

doi:10.1016/j.mtcomm.2020.101956

Cited by 6 publications

(10 citation statements)

References 28 publications

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“…In succeeding works, there would be another procedure (purely mathematical, semi analytical or fully numerical) that can be used for solving problems of space-time coefficients equation, where the coefficients are in more general forms and not in a variable separable form. This Time-space variable coefficients article specifically extends the previous works in Azis et al (2020Azis et al ( , 2021 for modified-Helmholtz and diffusion-reaction equations with time-independent coefficients to time-dependent coefficients modified-Helmholtz equation.…”

Section: Introductionsupporting

confidence: 55%

“…The computational study of several models for anisotropic media including functionally graded materials (FGMs) has attracted significant attention in recent years, and the development of these models has been a subject of intense research. Models with time-independent coefficients had been studied, for example, in AL-Bayati and Wrobel (2018), Ravnik and Tibuat (2019), Azis et al (2020, 2022), Abbaszadeh et al (2021, Azis (2022) and models with time-dependent coefficients, but with different aspects of study, in Luan and Khanh (2021), Geng et al (2021), Minh et al (2018), Hussein et al (2016), Van Bockstal and Slodi cka (2015), Lesnic et al (2013), Lesnic (2020, 2021), Sun et al (2019).…”

Section: Introductionmentioning

confidence: 99%

“…Models with time-independent coefficients had been studied, for example, in AL-Bayati and Wrobel (2018), Ravnik and Tibuat (2019), Azis et al . (2020, 2021, 2022), Abbaszadeh et al . (2021), Azis (2022) and models with time-dependent coefficients, but with different aspects of study, in Luan and Khanh (2021), Geng et al .…”

Section: Introductionmentioning

confidence: 99%

“…This article specifically extends the previous works in Azis et al . (2020, 2021) for modified-Helmholtz and diffusion-reaction equations with time-independent coefficients to time-dependent coefficients modified-Helmholtz equation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

Azis

2023

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PurposeTwo-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.Design/methodology/approach The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.FindingsThe main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.Originality/valueThe originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

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Section: Introductionsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

Azis

2023

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“…The relative displacement fundamental function is proposed based on the crack's peripheral equation. In [2], a combined Laplace transform and boundary element method is used to find numerical solutions to problems of another class of anisotropic functionally graded materials which are governed by a variable coefficients parabolic equation. A transformation is used to reduce the variable coefficients equation to a constant coefficients equation, which is then transformed into a boundary-only integral equation.…”

Section: Introductionmentioning

confidence: 99%

Applying a Set of Orthogonal Basis Functions in Numerical Solution of Hallen's Integral Equation for Dipole Antenna of Perfectly Conducting Material

Hatamzadeh

Masouri²

2022

J. Mod. Mater.

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A numerical solution for unsteady anisotropic diffusion reaction equation of time–space variable coefficients

Azis

2023

Math Methods in App Sciences

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Two‐dimensional transient problems for anisotropic functionally graded materials governed by a diffusion reaction equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time‐dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation, which can be written in forms of boundary‐only integral equations and solved using a standard boundary element method for the solutions. These boundary element method solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials are solved. The numerical solutions obtained are quite accurate.

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A boundary-only integral equation method for parabolic problems of another class of anisotropic functionally graded materials

Cited by 6 publications

References 28 publications

A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

Applying a Set of Orthogonal Basis Functions in Numerical Solution of Hallen's Integral Equation for Dipole Antenna of Perfectly Conducting Material

A numerical solution for unsteady anisotropic diffusion reaction equation of time–space variable coefficients

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