The Riccati System and a Diffusion-Type Equation

A Lie system is a system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a socalled Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the k-symplectic Lie systems, admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a k-symplectic structure. We devise new k-symplectic geometric methods to study their superposition rules, t-independent constants of motion and general properties. Our results are illustrated through examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of the k-symplectic geometry: systems of first-order ordinary differential equations.

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“…It was recently proved that diffusion equations and other PDEs can be approached through the Lie system [32,54] …”

Section: On the Need Of K-symplectic Lie Systemsmentioning

confidence: 99%

k-Symplectic Lie systems: theory and applications

Lucas

Vilariño

2015

Journal of Differential Equations

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“…Theoretically investigating the properties of systems in different contexts [1,2,3], such as for example classical and quantum physics [4,5,6,7,8,9,10,11,12,13,14,15,16], mathematics [17,18,19,20,21,22,23,24,25], biology [26,27], one is led to the consideration of the following non-linear non-autonomous first order differential equation…”

Section: Introductionmentioning

confidence: 99%

An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

et al. 2017

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A novel recipe for exactly solving in finite terms a class of special differential Riccati equations is reported. Our procedure is entirely based on a successful resolution strategy quite recently applied to quantum dynamical time-dependent SU(2) problems. The general integral of exemplary differential Riccati equations, not previously considered in the specialized literature, is explicitly determined to illustrate both mathematical usefulness and easiness of applicability of our proposed treatment. The possibility of exploiting the general integral of a given differential Riccati equation to solve an SU(2) quantum dynamical problem, is succinctly pointed out.

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“…A goal of this paper is to make a modest step in this direction (see also [32,53] and the references therein). We use explicit solutions from recent papers on variable quadratic Hamiltonians in nonrelativistic quantum mechanics [49,[54][55][56][57][58][59][60][61] to describe steady-state and transient solutions to linear cable equations derived for membrane compartments with a non-necessarily constant or monotonically changing radius and propose, en passage, a new hyperbolic representation for the neurite compartments.…”

Section: Introductionmentioning

confidence: 99%

A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

2014

Self Cite

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Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.

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The Riccati System and a Diffusion-Type Equation

Abstract: We discuss a method of constructing solution of the initial value problem for diffusiontype equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered.

Cited by 24 publications

References 49 publications

k-Symplectic Lie systems: theory and applications

k-Symplectic Lie systems: theory and applications

An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

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