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K. Krishnakumar

Publications and source records attributed to K. Krishnakumar.

7 recordsLinked to original sources
arXiv2020

Lie Symmetries and Similarity transformations for the Generalized Boiti-Leon-Pempinelli equations

We perform a detailed classification of the Lie point symmetries and of the resulting similarity transformations for the Generalized Boiti-Leon-Pempinelli equations. The latter equations for a system of two nonlinear 1+2 partial differential equations of second- and third-order. The nonlinear equations depend of two parameters, namely $n$ and $m$, from where we find that for various values of these two parameters the resulting systems admit different number of Lie point symmetries. For every case, we present the complete analysis for the admitted Lie group as also we determine all the possible similarity solutions which follow from the one-dimensional optimal system. Finally, we summarize the results by presenting them in a tabular way.

nlin.SI↗
arXiv2020

Lie symmetries and similarity solutions for the generalized Zakharov equations

The theory of Lie point symmetries is applied to study the generalized Zakharov system with two unknown parameters. The system reduces into a three-dimensional real value functions system, where we find that admits five Lie point symmetries. From the resulting point, we focus on these which provide travel-wave similarity transformation. The reduced system can be integrated while we remain with a system of two second-order nonlinear ordinary differential equations. The parameters of the latter system are classified in order the equations to admit Lie point symmetries. Exact travel-wave solutions are found, while the generalized Zakharov system can be described by the one-dimensional Ermakov-Pinney equation.

nlin.SI↗
arXiv2020

Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers-Fisher Equation

In this article, we study the Lie point symmetries for the time fractional generalized Burgers-Fisher (GBF) equation. While getting an appropriate combination of symmetries, the time fractional partial differential equation has been transformed to nonlinear fractional ordinary differential equations (ODE) using Erdelyi-Kober differential operator. Furthermore, using power series method, we get the exact solution of the nonlinear fractional GBF equation with the arbitrary nonlinearity.

nlin.SI↗
arXiv2020

Exact solutions of time fractional generalized Burgers-Fisher Equation using generalized Kudryashov method

In this article, we study the generalised Kudryashov method for the time fractional generalized Burgers-Fisher equation (GBF). Using traveling wave transformation, the time fractional GBF is transformed to nonlinear ordinary differential equation (ODE). Later, in the nonlinear ODE of timefractional GBF, the generalized Kudryashov and power series method is applied to get exact solutions.

nlin.SI↗
arXiv2015

Comment on "Classification of Cosmic Scale Factor via Noether Gauge Symmetries" [Int. J. Theor. Phys. 54, 2343 (2015)]

We discuss the relationship between the Noether point symmetries of the geodesic Lagrangian, in a (pseudo)Riemannian manifold, with the elements of the Homothetic algebra of the space. We observe that the classification problem of the Noether symmetries for the geodesic Lagrangian is equivalent with the classification of the Homothetic algebra of the space, which in the case of a Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime is a well-known result in the literature.

gr-qc↗
arXiv2015

Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility, $\sigma$, in which the last is defined by a stochastic differential equation with an Orstein--Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, $S$, and a new variable, $y$. We find that for arbitrary functional form of the volatility, $\sigma(y)$, the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when $\sigma(y)=\sigma_{0}$ and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black--Scholes--Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein--Stein model.

math.AP↗