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F. M. Mahomed

Publications and source records attributed to F. M. Mahomed.

At least 19 recordsLinked to original sources
arXiv2026

Invariant characterization of scalar third-order ODEs that admit the maximal contact symmetry Lie algebra

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits the maximal ten-dimensional contact symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently determine the contact transformation that does the reduction to the simplest linear equation $\bar{u}'''=0$. Furthermore, ample examples are given to illustrate our method.

math.CA↗
arXiv2026

Linearization Problem for a System of Two Second-Order ODEs via Cartan's Method: Branch I

Cartan's method classifies the class of linearizable system of two second-order ODEs into many branches. This paper investigates Branch I of the classification, characterized by a rank-one generalized Wilczynski invariant matrix and the vanishing of two relative invariants $K_1$ and $L_1$. It is demonstrated that any linearizable system belonging to this branch admits an eight-dimensional Lie point symmetry algebra. The canonical form for this class is provided and the invariant characterizations based on the obtained rank-zero invariant coframe and the corresponding constant structure equations are established. Also, a systematic procedure for constructing the linearizing point transformation is derived. The theoretical results are illustrated by several examples.

math.GM↗
arXiv2026

Linearization Problem for Third-Order ODEs with Four- and Five-Dimensional Lie Symmetry Algebras under Contact Transformations

Using Cartan equivalence method, invariant coframes are constructed for two branches of rank one and zero, which characterize linearizable third-order ODEs under contact transformations with four- and five-dimensional Lie symmetry algebras, respectively. A procedure for deriving the corresponding contact transformations is also presented, along with illustrative examples.

math.GM↗
arXiv2026

Equivalence Problem for Non-Linearizable Fourth-Order ODEs with Five-Dimensional Lie Symmetry subalgebra via Inductive Cartan Equivalence Method

Four coframes of invariant 1-forms are explicitly constructed using the Inductive Cartan equivalence method with rank zero corresponding to four distinct branches. These coframes are employed to characterize non-linearizable fourth-order ODEs under point transformation with a five-point symmetry Lie subalgebra. Moreover, we propose a procedure for obtaining the point transformation by using the derived invariant coframes, demonstrated through examples.

math.GM↗
arXiv2026

Equivalence Problem for Non-Linearizable Third-Order ODEs with Four-Dimensional Lie Symmetry Subalgebras under Point Transformations

Cartan's equivalence method is applied to explicitly construct invariant coframes for four branches, which are used to characterize all non-linearizable third-order ODEs with a four-dimensional Lie symmetry subalgebra under point transformations. Additionally, we present a method for constructing the point transformations based on the derived invariant coframes. Examples are provided to illustrate our approach.

math.GM↗
arXiv2025

Fractional Logistic Growth with Memory Effects: A Tool for Industry-Oriented Modeling

The logistic growth model is a classical framework for describing constrained growth phenomena, widely applied in areas such as population dynamics, epidemiology, and resource management. This study presents a generalized extension using Atangana-Baleanu in Caputo sense (ABC)-type fractional derivatives. Proportional time delay is also included, allowing the model to capture memory-dependent and nonlocal dynamics not addressed in classical formulations. Free parameters provide flexibility for modeling complex growth in industrial, medical, and social systems. The Hybrid Sumudu Variational (HSV) method is employed to efficiently obtain semi-analytical solutions. Results highlight the combined effects of fractional order and delay on system behavior. This approach demonstrates the novelty of integrating ABC-type derivatives, proportional delay, and HSV-based solutions for real-world applications.

eess.SY↗
arXiv2022

Symmetry Classification of Scalar $n$th Order Ordinary Differential Equations

We complete the Lie symmetry classification of scalar nth order, $n \geq 4$, ordinary differential equations by means of the symmetry Lie algebras they admit. It is known that there are three types of such equations depending upon the symmetry algebra they possess, viz. first-order equations which admit infinite dimensional Lie algebra of point symmetries, second-order equations possessing the maximum eight point symmetries and higher-order, $n \geq 3$, admitting the maximum $n + 4$ dimensional symmetry algebra. We show that nth order equations for $n \geq 4$ do not admit maximally an $n + 3$ dimensional Lie algebra except for $n = 5$ which can admit $sl(3, R)$ algebra. Also, they can possess an $n + 2$ dimensional Lie algebra that gives rise to a nonlinear equation that is not linearizable via a point transformation. It is shown that for $n \geq 5$ there is only one such class of equations.

math-ph↗
arXiv2018

Linearization of third-order ordinary differential equations u'''=f(x,u,u',u'') via point transformations

The linearization problem by use of the Cartan equivalence method for scalar third-order ODEs via point transformations was solved partially in [1,2]. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation u'''=f(x,u,u',u'') which admits a four-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The method provides auxiliary functions which can be utilized to efficiently determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.

math.CA↗
arXiv2017

Invariant characterization of scalar third-order ODEs that admit the maximal point symmetry Lie algebra

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'"=f(x,u,u',u")$ which admits the maximal seven-dimensional point symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation $\bar{u}'"=0$. Moreover, examples are given to illustrate the method.

math.CA↗
arXiv2017

Invariant characterization of third-order ODEs $u'''=f(x,u,u',u'')$ that admit a five-dimensional point symmetry Lie algebra

The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function $f$ in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.

math.CA↗
arXiv2015

Invariants for systems of two linear hyperbolic-type equations by complex methods

Invariants of general linear system of two hyperbolic partial differential equations (PDEs) are derived under transformations of the dependent and independent variables by real infinitesimal method earlier. Here a subclass of the general system of linear hyperbolic PDEs is investigated for the associated invariants, by complex as well as real methods. The complex procedure relies on the correspondence of systems of PDEs with the base complex equation. Complex invariants of the base complex PDEs are shown to reveal invariants of the corresponding systems. A comparison of all the invariant quantities obtained by complex and real methods for this class, is presented which shows that the complex procedure provides a few invariants different from those extracted by real symmetry analysis.

math.CA↗
arXiv2014

Invariants of third-order ordinary differential equations $y'''=f(x,y,y',y'')$ via fiber preserving transformations

Bagderina \cite{Bagderina2008} solved the equivalence problem for scalar third-order ordinary differential equations (ODEs), quadratic in the second-order derivative, via point transformations. However, the question is open for the general class $y'''=f(x,y,y',y'')$ which is not quadratic in the second-order derivative. We utilize Lie's infinitesimal method to study the differential invariants of this general class under pseudo-group of fiber preserving equivalence transformations $\bar{x}=ϕ(x), \bar{y}=ψ(x,y)$. As a result, all third-order differential invariants of this group and the invariant differentiation operators are determined. This leads to simple necessary explicit conditions for a third-order ODE to be equivalent to the respective canonical form under the considered group of transformations. Applications motivated by the literature are presented.

math.CA↗
arXiv2013

A Partial Hamiltonian Approach for Current Value Hamiltonian Systems

We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.

math.OC↗
arXiv2013

$λ$-symmetry criteria for linearization of second order ODEs via point transformations

An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $λ$-symmetries and first integrals. This relation further leads to a new $λ$-symmetry linearization criteria for second order ODEs which provides a new approach for constructing the linearization transformations with lower complexity. The effectiveness of the approach is illustrated by obtaining the local linearization transformations for the linearizable nonlinear ODEs of the form $y''+F_1(x,y)y'+F(x,y)=0$. Examples of linearizing nonlinear ODEs which are quadratic or cubic in the first derivative are also presented.

math.CA↗
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

Noether Symmetry Approach in f(R) Tachyon Model

In this Letter by utilizing the Noether symmetry approach in cosmology, we attempt to find the tachyon potential via the application of this kind of symmetry to a flat Friedmann-Robertson-Walker (FRW) metric. We reduce the system of equations to simpler ones and obtain the general class of the tachyon's potential function and $f(R)$ functions. We have found that the Noether symmetric model results in a power law $f(R)$ and an inverse fourth power potential for the tachyonic field. Further we investigate numerically the cosmological evolution of our model and show explicitly the behavior of the equation of state crossing the cosmological constant boundary.

physics.gen-ph↗
arXiv2011

Noether Gauge Symmetry Approach in f(R) Gravity

We discuss the f(R) gravity model in which the origin of dark energy is identified as a modification of gravity. The Noether symmetry with gauge term is investigated for the f(R) cosmological model. By utilization of the Noether Gauge Symmetry (NGS) approach, we obtain two exact forms f(R) for which such symmetries exist. Further it is shown that these forms of f(R) are stable.

physics.gen-ph↗