Lie Symmetries of Fokker-Planck Equations with Logarithmic diffusion and Drift Terms

Silva, Érica de Mello; Filho, T. M. Rocha; Santana, A. E.

doi:10.1088/1742-6596/40/1/019

Cited by 13 publications

(12 citation statements)

References 16 publications

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“…The constraint on the range of the stochastic variable x suggests that the cells do not grow forever in size; in other words, there exists an upper bound on the size of the cells. This bounded stochastic process, which was first presented by Pesz [16] and then further generalized by Cardeal et al [6] and Silva et al [21], has found applications in various systems with spatial confinement, e.g. a Brownian walker trapped between fixed plates (see Ref.…”

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confidence: 95%

A Modified Stochastic Gompertz Model for Tumour Cell Growth

2008

Computational and Mathematical Methods in Medicine

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Based upon the deterministic Gompertz law of cell growth, we have proposed a stochastic model of tumour cell growth, in which the size of the tumour cells is bounded. The model takes account of both cell fission (which is an ‘action at a distance’ effect) and mortality too. Accordingly, the density function of the size of the tumour cells obeys a functional Fokker–Planck Equation (FPE) associated with the bounded stochastic process. We apply the Lie-algebraic method to derive the exact analytical solution via an iterative approach. It is found that the density function exhibits an interesting kink-like structure generated by cell fission as time evolves.

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confidence: 95%

A Modified Stochastic Gompertz Model for Tumour Cell Growth

2008

Computational and Mathematical Methods in Medicine

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“…This procedure has been adopted to derive, for instance, generalized Fokker-Planck equations from a given symmetry of a reaction-diffusion equation [37]. We followed along this line to study and classify some generalized Kðm; nÞ equations by choosing the classical KdV equation Lie-symmetry algebra.…”

Section: Introductionmentioning

confidence: 99%

Time-dependent exact solutions for Rosenau–Hyman equations with variable coefficients

Souza

Silva

2015

Communications in Nonlinear Science and Numerical Simulation

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“…Our package is well suited for efficiently computing Lie symmetries of large systems, as for instance the Yang-Mills with SU(2) and SU(3) gauge group [26], and has been used in the last years in our group in different applications [27,28,29,30,31].…”

Section: Introductionmentioning

confidence: 99%

[SADE] a Maple package for the symmetry analysis of differential equations

Filho

Figueiredo

2011

Computer Physics Communications

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We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, non classical, Lie-Bäcklund and potential symmetries, invariant solutions, first-integrals, Nöther theorem for both discrete and continuous systems, solution of ordinary differential equations, order and dimension reductions using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented by the authors in the package QPSI) for the determination of quasipolynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given. Nature of the physical problem: Determination of analytical properties of systems of differential equations, including symmetry transformations, analytical solutions and conservation laws. Method of resolution:The package implements in MAPLE some algorithms (discussed in the text) for the study of systems of differential equations.Restrictions on the complexity of the problem: Depends strongly on the system and on the algorithm required. Typical restrictions are related to the solution of a large over-determined system of linear or non-linear differential equations.Typical running time: Depends strongly on the order, the complexity of the differential system and the object computed. Ranges from seconds to hours.

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Lie Symmetries of Fokker-Planck Equations with Logarithmic diffusion and Drift Terms

Cited by 13 publications

References 16 publications

A Modified Stochastic Gompertz Model for Tumour Cell Growth

A Modified Stochastic Gompertz Model for Tumour Cell Growth

Time-dependent exact solutions for Rosenau–Hyman equations with variable coefficients

[SADE] a Maple package for the symmetry analysis of differential equations

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