Complete Symmetry Group Research Articles

Application of Lie symmetries to solving modified Black-Scholes equation

We perform Lie symmetry analysis on the modified Black-Scholes model described as a partial differential equation (PDE). As a result, a new complete Lie symmetry group and infinitesimal generators of the one-dimensional modified fractional Black-Scholes model are derived. Furthermore, we compute a family of exact invariant solutions that constitute the modified fractional Black-Scholes model using the associated infinitesimal generators and the corresponding similarity reduction equations. Using known solutions, more solutions are generated via group point transformations.

Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis

Modern weather and climate prediction models are based on a system of nonlinear partial differential equations called the primitive equations. Lie symmetries of the primitive equations with zero external heating rate are computed and the structure of its maximal Lie invariance algebra, which is infinite-dimensional, is studied. The maximal Lie invariance algebra for the case of a nonzero constant Coriolis parameter is mapped to the case of vanishing Coriolis force. The same mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame. This mapping is used to obtain exact solutions for the rotating case from exact solutions for the nonrotating equations. Another important result of the paper is the computation of the complete point symmetry group of the primitive equations using the algebraic method.

Symmetries and special solutions of a parabolic chemotaxis system

In this paper, we consider a class of chemotaxis models with two arbitrary constitutive functions g(u) and f(v). After having performed a complete symmetry group classification with respect to them the reduced systems are derived. By considering g(u) of the logistic form, wide classes of exact solutions are found.

Extended symmetry analysis of an isothermal no-slip drift flux model

We perform extended group analysis for a system of differential equations modeling an isothermal no-slip drift flux. The maximal Lie invariance algebra of this system is proved to be infinite-dimensional. We also find the complete point symmetry group of this system, including discrete symmetries, using the megaideal-based version of the algebraic method. Optimal lists of one- and two-dimensional subalgebras of the maximal Lie invariance algebra in question are constructed and employed for obtaining reductions of the system under study. Since this system contains a subsystem of two equations that involves only two of three dependent variables, we also perform group analysis of this subsystem. The latter can be linearized by a composition of a fiber-preserving point transformation with a two-dimensional hodograph transformation to the Klein–Gordon equation. We also employ both the linearization and the generalized hodograph method for constructing the general solution of the entire system under study. We find inter alia genuinely generalized symmetries for this system and present the connection between them and the Lie symmetries of the subsystem we mentioned earlier. Hydrodynamic conservation laws and their generalizations are also constructed.

Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System

Abstract A complete symmetry group classification for the system of shallow water equations with the horizontal temperature gradient, also known as Ripa system, is presented. A rigorous and systematic procedure based on the general invariants of the adjoint representation is used to construct the one-dimensional optimal system of the Lie algebra. The complete inequivalence class of the group invariant solutions are obtained by using the one-dimensional optimal system. One such solution of the Ripa system is used to study the evolutionary behaviour of the discontinuity wave.

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier-Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

Equivalence and symmetries for variable coefficient linear heat type equations. I

A systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations (PDEs) is presented. The complete symmetry group classification is re-performed in the current context. A useful criterion which is necessary and sufficient for being reducible to the standard heat equation by point transformations is established. A similar criterion is also valid for the equations to have a four- or six-dimensional symmetry group (nontrivial symmetry groups). In this situation, the basis elements are listed in terms of coefficients. A number of illustrative examples are given. In particular, some applications from the recent literature are re-examined in our new approach. Multidimensional parabolic PDEs of heat and Schrödinger type are also considered.

Coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry: Symmetries and conservation laws

A complete symmetry group classification of a coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is carried out. This system is a natural two-components generalization of the Klein–Gordon–Fock system with central symmetry. Moreover, conservation laws for the system are obtained. Some interesting physical conclusions related to the derived conserved vectors are discussed.

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

In this paper we make a Lie symmetry analysis of a generalized nonlinear beam equation with both second-order and fourth-order wave terms, which is extended from the classical beam equation arising in the historical events of travelling wave behavior in the Golden Gate Bridge in San Francisco. We perform a complete Lie symmetry group classification by using the equivalence transformation group theory for the equation under consideration. Lie symmetry reductions of a nonlinear beam-like equation which are singled out from the classification results are investigated. Some classes of exact solutions, including solitary wave solutions, triangular periodic wave solutions and rational solutions of the nonlinear beam-like equations are constructed by means of the reductions and symbolic computation.

SYMMETRY PROPERTIES OF A NOVEL RICCATI SEQUENCE

Ponte Academic JournalNov 2017, Volume 73, Issue 11 SYMMETRY PROPERTIES OF A NOVEL RICCATI SEQUENCEAuthor(s): Adhir Maharaj ,DB Lortan, M. GovenderJ. Ponte - Nov 2017 - Volume 73 - Issue 11 doi: 10.21506/j.ponte.2017.11.35 Abstract:Recursion operators and hierarchies of evolution partial differential equations have been well-established for the past forty years. An algorithm to calculate recursion operators may be found in \cite{Bluman a}. Recently sequences of ordinary differential equations were introduced which were investigated in terms of it's symmetry and singularity properties. We analyse the sequence in this paper in terms of it's Lie point symmetries and the possession of a complete symmetry group. Download full text:Check if you have access through your login credentials or your institution Username Password

A Differential Sequence Generated By A Second-Order Recursion Operator

Ponte Academic JournalOct 2017, Volume 73, Issue 10 A Differential Sequence Generated By A Second-Order Recursion OperatorAuthor(s): Adhir MaharajJ. Ponte - Oct 2017 - Volume 73 - Issue 10 doi: 10.21506/j.ponte.2017.10.3 Abstract:The motivation of our investigation is the paper by Petersson {\it et al} \cite{Petersson a} where recursion operators were obtained for a class of semilinear and quasilinear evolution equations. We consider the class of semilinear evolution equations as the basis of our investigation. By setting $a=0, b=0, c=0$ and $u_t=0$ we obtain an ordinary differential equation. We analyse the resulting differential sequence in terms of it's symmetry properties. We also obtain the complete symmetry group for each element of the sequence. Download full text:Check if you have access through your login credentials or your institution Username Password

Putting Queens in Carry Chains, N o̱27

The N-Queens Puzzle is a fascinating combinatorial problem. Up to now, the number of distinct valid placements of N non-attacking queens on a generalized N × N chessboard cannot be computed by a formula. The computation of these numbers is instead based on an exhaustive search whose complexity grows dramatically with the problem size N. Solutions counts are currently known for all N up to 26. The parallelization of the search for solutions is embarrassingly simple. It is achieved by pre-placing the queens within a certain board region. These pre-placements partition the search space. The chosen extent of the pre-placement allows for a wide range of the partitioning granularity. This ease of partitioning makes the N-Queens Puzzle a great show-off case for tremendously parallel computation approaches and a flexible benchmark for parallel compute resources. This article presents the Q27 Project, an open-source effort targeting the computation of the solution count of the 27-Queens Puzzle. It is the first undertaking pushing the frontier of the N-Queens Puzzle that exploits the complete symmetry group D4 of the square. This reduces the overall computational complexity already to an eighth in comparison to a naive exploration of the whole search space. This article details the coronal pre-placement that enables the partitioning of the overall search under this approach. With respect to the physical implementation of the computation, it describes the hardware structure that explores the resulting subproblems efficiently by exploiting bit-level operations and their mapping to FPGA devices as well as the infrastructure that organizes the contributing devices in a distributed computation. The performance of several FPGA platforms is evaluated using the Q27 computation as a benchmark, and some intriguing observations obtained from the available partial solutions are presented. Finally, an estimate on the remaining run time and on the expected magnitude of the final result is dared.

Nest graphs and minimal complete symmetry groups for magic Sudoku variants

A symmetry group for Sudoku is complete if its action partitions the set of Sudoku boards into all possible orbits, and minimal if no group of smaller size would do the same. Previously, for a 4 x 4 Sudoku variation known as Shidoku, the authors used an analogous symmetry group to partition the set of Shidoku boards into so-called nests and then use the interplay between the physical and relabeling symmetries to find certain subgroups that were both complete and minimal. In this paper these same techniques are applied to find a minimal complete symmetry group for the modular magic Sudoku variation, as well as for another Sudoku variation called semi-magic Sudoku. The paper concludes with a simple computation which leads to the non-obvious fact that the full Sudoku symmetry group is, in fact, already minimal and complete.

Algebraic method for finding equivalence groups

The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the corresponding equivalence algebra. Two versions of the method are presented, where the first involves the automorphism group of this algebra and the second is based on a list of its megaideals. We illustrate the megaideal-based version of the method with the computation of the complete equivalence group of a class of nonlinear wave equations with applications in nonlinear elasticity.

Optimal system, symmetry reductions and new closed form solutions for the geometric average Asian options

In this paper, the Lie group analysis method is applied to the geometric average Asian option pricing equation in financial problems. Firstly, the complete Lie symmetry group and infinitesimal generators of this equation are derived. Then the optimal system with one parameter for the Lie symmetry algebra are obtained, which gives the possibility to describe a complete set of invariant solutions to the pricing equation. Finally, based on the optimal system the symmetry reductions and corresponding closed form solutions for the pricing equation are proposed.

Undefined Jacobi last multiplier? Complete symmetry group!

The Jacobi last multiplier has an intimate association with the symmetries of differential equations. We present four examples in which the inability to obtain a last multiplier using the method of Lie itself provides us with information. We show that the three representations of the complete symmetry group of the linear harmonic oscillator can be obtained by searching for Jacobi last multipliers: they correspond to the Lie point symmetries that have zero determinant, thus far regarded as a useless case. The point is emphasized using the examples of the Volterra–Verhulst–Pearl equation, the Kepler problem and a scalar equation of fourth order.

Complete point symmetry group of the barotropic vorticity equation on a rotating sphere

The complete point symmetry group of the barotropic vorticity equation on a rotating sphere is determined. The method we use relies on the invariance of megaideals of the maximal Lie invariance algebra of a system of differential equations under automorphisms generated by the associated point symmetry group. A convenient set of megaideals is found for the maximal Lie invariance algebra of the spherical vorticity equation. We prove that there are only two independent (up to composition with continuous point symmetry transformations) discrete symmetries for this equation.

Generalized Runge-Lenz vector and a hidden symmetry of the nine-dimensional MICZ-Kepler problem

The problem of a particle in the Coulomb potential in a nine-dimensional space with the presence of the SO(8) monopole has been investigated recently and is called the nine-dimensional MICZ-Kepler problem. In this paper, we construct a new integral of motion of the system and name it the generalized Runge-Lenz vector. This integral of motion relates to a hidden symmetry of the problem, and hence we are able to construct explicitly a complete symmetry group of the problem, which is SO(10) instead of SO(9).

Invariant boundary value problems for a fourth-order dynamic Euler-Bernoulli beam equation

We obtain the complete Lie symmetry group classification of the dynamic fourth-order Euler-Bernoulli partial differential equation, where the elastic modulus, the area moment of inertia are constants and the applied load is a nonlinear function. In the Lie analysis, the principal Lie algebra which is two-dimensional extends in three cases, viz., the linear, the exponential, and the general power law. For each of the nontrivial cases, we determine symmetry reductions to ordinary differential equations which are of order four. In only one case related to the power law we are able to have a compatible initial-boundary value problem for a clamped end and a free beam. For these cases we deduce the corresponding fourth-order ordinary differential equations with appropriate boundary conditions. We provide an asymptotic solution for the reduced fourth-order ordinary differential equation corresponding to a clamped or free beam.