Lie symmetry analysis of two dimensional weakly singular integral equations

Pashayi, S.; Shahmorad, S.; Hashemi, Mir Sajjad

doi:10.1016/j.geomphys.2021.104385

Cited by 6 publications

(2 citation statements)

References 29 publications

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“…Importantly, although not explicitly stated, each cell population might have the ability to diffuse more rapidly in regions of white matter. Lie group methods [8][9][10][11][12][13][14][15][16][17] possess a versatile toolkit with broad applications in the realms of science and mathematics. They provide a systematic approach to understanding symmetries, solving differential equations, and gaining insights into complex physical and mathematical systems.…”

Section: Introductionmentioning

confidence: 99%

Invariant analysis of the two-cell tumor growth model in the brain

Hussain,

Usman,

Zaman

2024

Phys. Scr.

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In this study, we explore the invariance properties of a tumor growth model involving two distinct cell populations. These populations are characterized by different diffusion coefficients but share a common killing rate. This particular model serves as a representation of tumor growth within the brain. By employing the Lie group method, we unveil a two-dimensional symmetry algebra for cases where both diffusion coefficients are allowed to vary arbitrarily. Interestingly, this method reveals a nine-dimensional symmetry algebra when the diffusion coefficients are held constant. In both scenarios involving varying and constant diffusion coefficients, we conduct similarity reductions to deduce group invariant solutions, thus elucidating the model's behavior. Notably, our findings demonstrate that the tumor's growth remains exponential irrespective of the presence or absence of a killing rate. This remarkable phenomenon holds for various configurations of diffusion coefficients. To validate our observations, we employ Mathematica simulations, which corroborate the model's exponential growth behavior and emphasize the role of killing rates, diffusion coefficients, and growth rate parameters in driving this exponential trend. Also, the conserved flows and conserved quantities of the model are demonstrated.

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Section: Introductionmentioning

confidence: 99%

Invariant analysis of the two-cell tumor growth model in the brain

Hussain,

Usman,

Zaman

2024

Phys. Scr.

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“…For example, Levi and Rodriguez successfully extended this method to the case of difference equations [25]. Again, the μsymmetry reduction method is used to deal with partial differential equations (PDEs) [26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

New symmetry reduction method for (1+1)-dimensional differential-difference equations

Lyu¹,

Shi

Ren

et al. 2023

Front. Phys.

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We propose a new symmetry reduction method for (1+1)-dimensional differential-difference equations (DDEs), namely, the λ-symmetry reduction method of solving ordinary differential equations is generalized to DDEs. Order-reduction processes are a consequence of the invariance of the given DDE under vector fields of the new class. These vector fields satisfy a new prolongation formula. A simple example of order-reduction is provided to illustrate the application.

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