scholarly journals On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Winter Sinkala
Keyword(s):  
Exact Solutions ◽  
Equivalence Transformation ◽  
Equivalent Transformation ◽  
Invariant Solutions ◽  
Governing Equations ◽  
New Exact Solutions ◽  
Classical Symmetries ◽  
Black Scholes ◽  
Nonclassical Symmetries ◽  
The Given

The nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. Through this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. The challenge with the nonclassical symmetries method, however, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and therefore difficult to solve. In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. This is what we illustrate in this paper. We construct four nontrivial nonclassical symmetries of the Black–Scholes equation by transforming nonclassical symmetries of the heat equation. For completeness, we also construct invariant solutions of the Black–Scholes equation associated with the determined nonclassical symmetries.

scholarly journals Symmetry Reduction, Exact Solutions, and Conservation Laws of the ZK-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Wenbin Zhang ◽  
Jiangbo Zhou ◽  
Sunil Kumar
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Similarity Reduction ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Reduction Group ◽  
The Given ◽  
Bbm Equation

Employing the classical Lie method, we obtain the symmetries of the ZK-BBM equation. Applying the given Lie symmetry, we obtain the similarity reduction, group invariant solution, and new exact solutions. We also obtain the conservation laws of ZK-BBM equation of the corresponding Lie symmetry.

scholarly journals Exact Solutions of Boundary Layer Equations in Polymer Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2101
Author(s):  
Oksana A. Burmistrova ◽  
Sergey V. Meleshko ◽  
Vladislav V. Pukhnachev
Keyword(s):  
Boundary Layer ◽  
Exact Solutions ◽  
Invariant Solution ◽  
Stationary Problem ◽  
Solid Boundary ◽  
Invariant Solutions ◽  
Boundary Layer Equations ◽  
New Exact Solutions ◽  
Logarithmic Curve ◽  
Solutions Of Equations

The paper presents new exact solutions of equations derived earlier. Three of them describe unsteady motions of a polymer solution near the stagnation point. A class of partially invariant solutions with a wide functional arbitrariness is found. An invariant solution of the stationary problem in which the solid boundary is a logarithmic curve is constructed.

scholarly journals Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 527 ◽  
Author(s):  
Adem Kilicman ◽  
Rathinavel Silambarasan
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Complex Structures ◽  
Algebraic Equations ◽  
New Exact Solutions ◽  
Modified Kudryashov Method ◽  
Kudryashov Method ◽  
The Given ◽  
Sivashinsky Equation

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems yields the analytical solution. By this way, various exact solutions including complex structures are found, and their behavior is drawn in the 2D plane by Maple to compare the uniqueness and wave traveling of the solutions.

New soliton solutions of conformable time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation in modeling wave phenomena

2019 ◽  
Vol 33 (18) ◽  
pp. 1950202 ◽  
Author(s):  
S. Saha Ray
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Salient Feature ◽  
Wave Transformation ◽  
Composite Function ◽  
New Exact Solutions ◽  
The Given ◽  
Wave Phenomena

In this paper, new traveling wave solutions of time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation have been revealed by the help of extended Kudryashov’s method. In this case, the time fractional derivative has been considered in the sense of conformable fractional derivative. The salient feature of the proposed extended Kudryashov’s method is to take full advantage of solutions for the Riccati and the Bernoulli equations involving parameters. Using some particular wave transformation, the given equation is converted to the nonlinear ODE by the help of chain rule and the derivative of a composite function; and then, a novel analytical method viz. extended Kudryashov’s method has been used to solve the resulting equation. Consequently, some new exact solutions have been successfully obtained. The exact solutions devised by the proposed method manifest that the proposed method is effective and easy to implement.

Analytic study of solutions for a two-component Novikov system

2020 ◽  
pp. 2150025
Author(s):  
Hui Gao ◽  
Gangwei Wang
Keyword(s):  
Exact Solutions ◽  
Stationary Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Two Component ◽  
Group Invariant Solutions ◽  
Symmetry Transformations ◽  
Symmetry Method ◽  
Similarity Reductions

Under investigation in this paper is a two-component Novikov system (also called Geng-Xue equation), which was proposed by Geng and Xue in 2009. Firstly, via the Lie symmetry method, infinitesimal generators, commutator table of Lie algebra and symmetry groups of the two-component Novikov system are presented. At the same time, some group invariant solutions are computed through similarity reductions. In particular, we construct peakon solution by applying the distribution theory. In addition, based on obtained group invariant solutions and symmetry transformations, we derive some new exact solutions, which include stationary solutions, smooth solutions, and a weak solution. The analytical properties to some of group invariant solutions and new exact solutions are discussed, such as decay, asymptotic behavior, and boundedness.

scholarly journals A NEW CONSTRUCTION OF OS OF SUBALGEBRAS AND INVARIANT SOLUTION OF THE BLACK-SCHOLES EQUATION

2021 ◽  
Vol 16 (9) ◽  
Author(s):  
Zahid Hussain
Keyword(s):  
Differential Equation ◽  
Lie Group ◽  
Invariant Solution ◽  
Group Analysis ◽  
Invariant Solutions ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Group Invariant Solutions ◽  
New Construction ◽  
The Given

In this manuscript, the Lie group technique is applied to construct a new OS and invariant solutions of a one-dimensional LA, which describes the symmetries properties of a nonlinear Black-Scholes model. The structure of LA depends on one parameter. We have shown a novel way to construct the so-called OS of subalgebras of the Black-Scholes equation by utilizing the given symmetries. We transform the symmetries of the Black-Scholes equation into a simple ordinary differential equation called the Lie equation, which provides us a way through which to construct a new optimal scheme of subalgebras of the Black-Scholes through applying the concept of LE. The OS which consists of minimal representatives is utilized to develop the invariant solution for the Black-Scholes equation. The fundamental use of the Lie group analysis to the differential equation is the categorization of group invariant solutions of differential equations via OS. Finally, we have utilized the OS to construct the invariant solution of the Black-Scholes equation.

New Exact Solutions of Steady Plane Flows of an In Compressible Fluid of Variable Viscosity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Sobia Younus
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Compressible Fluid ◽  
Variable Viscosity ◽  
Plane Motion ◽  
Transformation Technique ◽  
Vorticity Distribution ◽  
New Exact Solutions ◽  
Steady Plane ◽  
Uniform Stream

<span>Some new exact solutions to the equations governing the steady plane motion of an in compressible<span> fluid of variable viscosity for the chosen form of the vorticity distribution are determined by using<span> transformation technique. In this case the vorticity distribution is proportional to the stream function<span> perturbed by the product of a uniform stream and an exponential stream<br /><br class="Apple-interchange-newline" /></span></span></span></span>

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

Sign in / Sign up

Export Citation Format

Share Document