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J. C. Ndogmo

Publications and source records attributed to J. C. Ndogmo.

At least 19 recordsLinked to original sources
arXiv2023

Coefficient characterization of linear differential equations with maximal symmetries

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its general form such an equation can be expressed in terms of only two arbitrary functions, and its connection with the Laguerre-Forsyth form is clarified. The characterizing conditions are also used to derive an infinite family of semi-invariants, each corresponding to an arbitrary order of the linear equation. Finally a simplifying ansatz is established, which allows an easier determination of the infinitesimal generators of the induced pseudo group of equivalence transformations, for all the three most general canonical forms of the equation.

math.CA↗
arXiv2022

Equivalence classes and Linearization of the Riccati and Abel chain

The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from the chain are not linearizable by any point transformations. The Lie pseudo-group of equivalence transformations for equations of arbitrary orders from the chain are then found, together with expressions for the transformed parameter functions. An important subgroup of the group of equivalence transformations found is considered and some associated equivalence classes are exhibited.

math.AP↗
arXiv2022

Generalization of non-Cartan Symmetries to arbitrary dimensions

Second order scalar ordinary differential equations ({\sc ode}s) which are linearizable possess special types of symmetries. These are the only symmetries which are non fiber-preserving in the linearized form of the equation, and they are called non-Cartan symmetries and known only for scalar {\sc ode}s. We give explicit expressions of non-Cartan symmetries for systems of {\sc ode}s of arbitrary dimensions and show that they form an abelian Lie algebra. It is however shown that the natural extension of these non-Cartan symmetries to arbitrary dimensions is applicable only to the natural extension of scalar second order equations to higher dimensions, that is, to equivalence classes under point transformations of the trivial vector equation. More precisely, it is shown that non-Cartan symmetries characterize linear systems of {\sc ode}s reducible by point transformation to their trivial counterpart, and we verify that they do not characterize nonlinear systems of {\sc ode}s having this property. It is also shown amongst others that the non-Cartan property of a symmetry vector is coordinate-free. Some examples of application of these results are discussed. %

math.DG↗
arXiv2022

Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form

Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.

math.CA↗
arXiv2022

Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations

math.DG↗
arXiv2022

Group classification and exact solutions of a class of nonlinear waves

We apply an extension of a new method of group classification to a family of nonlinear wave equations labelled by two arbitrary functions, each depending on its own argument. The results obtained confirm the efficiency of the proposed method for group classification, termed the method of indeterminates. A model equation from the classified family of fourth order Lagrange equations is singled out. Travelling wave solutions of the latter are found through a similarity reduction by variational symmetry operators, followed by a double order reduction into a second order ordinary differential equation. Multi-soliton solutions and other exact solutions are also found by various methods including Lie group and Hirota methods. The most general action of the full symmetry group on any given solution is provided. Some remarkable facts on Lagrange equations emerging from the whole study are outlined.

math.AP↗
arXiv2022

Integrable classes of a family of evolution equations

The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the results are transformed into known integrable equations from the literature. Recursion operators are also given for all the symmetry integrable classes found, some of which are the only ones known for some of the canonical classes.

nlin.SI↗
arXiv2020

Lie group classification and conservation laws of a class of hyperbolic equations

A new method for the Lie group classification of differential equations is proposed. It is based of the determination of all possible cases of linear dependence of certain indeterminate appearing in the determining equations of symmetries of the equation. The method is simple and systematic and applied to a family of hyperbolic equations. Moreover, as the said family contains several known equations with important physical applications, low-order conservation laws of some relevant equations from the family are computed, and the results obtained are discussed with regard to the symmetry integrability of a particular class from the underlying family of hyperbolic equations.

math.AP↗
arXiv2016

Algebraic structure and maximal dimension of the symmetry algebra for arbitrary systems of ODEs

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation $\mathbf{y}^{(n)}$=0. For arbitrary systems of ordinary differential equations of order $n \geq 3$ reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

math.CA↗
arXiv2015

Group-invariant solutions of a nonlinear acoustics model

Based on a recent classification of subalgebras of the symmetry algebra of the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation into ordinary differential equations are obtained. Large classes of group-invariant solutions of the equation are also determined, and some properties of the reduced equations and exact solutions are discussed.

math.AP↗
arXiv2012

A characterization of iterative equations by their coefficients

An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. In particular, a simple necessary and sufficient condition for an equation to be iterative is given for the general fourth-order linear equation solely in terms of its coefficients.

math.CA↗
arXiv2011

On dummy variables of structure-preserving transformations

A method is given for obtaining equivalence subgroups of a family of differential equations from the equivalence group of simpler equations of a similar form, but in which the arbitrary functions specifying the family element depend on fewer variables. Examples of applications to classical equations are presented, some of which show how the method can actually be used for a much easier determination of the equivalence group itself.

math.AP↗
arXiv2011

Some comparison results on equivalence groups

This paper deals with the comparison of two common types of equivalence groups of differential equations, and this gives rise to a number of results presented in the form of theorems. It is shown in particular that one type can be identified with a subgroup of the other type. Consequences of this comparison related in particular to the determination of invariant functions of the differential equations are also discussed.

math.DG↗
arXiv2009

Group Classification of a family of second-order differential equations

We find the group of equivalence transformations for equations of the form $y''= A(x)y' + F(y),$ where $A$ and $F$ are arbitrary functions. We then give a complete group classification of these families of equations, using a direct method of analysis, together with the equivalence transformations.

math.AP↗
arXiv2008

Classification of barrier options

For a given level of accuracy in option prices, the paper considers the problem of deciding when exactly, as one or more of the pricing parameters change, a barrier option degenerates into a simpler type of option. This problem is meaningful in the real world where option prices are always determined within a certain level of accuracy. The problem is reduced to finding certain critical values of the initial stock price, and this is achieved through a probability-based approach.

q-fin.PR↗
arXiv2008

On generating relative and absolute invariants of linear differential equations

A general expression for a relative invariant of a linear ordinary differential equations is given in terms of the fundamental semi-invariant and an absolute invariant. This result is used to established a number of properties of relative invariants, and it is explicitly shown how to generate fundamental sets of relative and absolute invariants of all orders for the general linear equation. Explicit constructions are made for the linear ODE of order five. The approach used for the explicit determination of invariants is based on an infinitesimal method.

math.AP↗
arXiv2008

Invariants associated with linear ordinary differential equations

We apply a novel method for the equivalence group and its infinitesimal generators to the investigation of invariants of linear ordinary differential equations. First, a comparative study of this method is illustrated by an example. Next, the method is used to obtain the invariants of low order linear ordinary differential equations, and the structure invariance group for an arbitrary order of these equations. Other properties of these equations are also discussed, including the exact number of their invariants.

math.AP↗