In this paper, by applying Valenti’s theory for the approximate symmetry, we introduce and define... more In this paper, by applying Valenti’s theory for the approximate symmetry, we introduce and define the concept of a one-dimensional optimal system of approximate subalgebras for a generalized Ames’s equation; furthermore, the algebraic structure of the approximate Lie algebra is discussed. New approximately invariant solutions to the equation are found.
In this paper, we study an integrable system of coupled KdV equations, derived by Gear and Grimsh... more In this paper, we study an integrable system of coupled KdV equations, derived by Gear and Grimshaw (Stud. Appl. Math. 70(3):235-258, 1984), modeling the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratified fluid. In particular, we present the complete group classification of the model and find conditions on arbitrary parameters for which the system admits symmetries. Some exact solutions of physical relevance are derived.
Invariant solutions of partial differential equations are found by solving a reduced system invol... more Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.
A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with... more A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with a nonlinear source term is considered. The authors determine the Lie symmetries and reduce the original fractional partial differential equation into a fractional ordinary differential equation (FODE), in some interesting cases. Numerical solutions of FODEs are obtained, and through Lie transformations, numerical solutions of original equation are found.
A new approach, combining the Ibragimov method and the one by Anco and Bluman, with the aim of al... more A new approach, combining the Ibragimov method and the one by Anco and Bluman, with the aim of algorithmically computing local conservation laws of partial differential equations, is discussed. Some examples of the application of the procedure are given. The method, of course, is able to recover all the conservation laws found by using the direct method; at the same time we can characterize which symmetry, if any, is responsible for the existence of a given conservation law. Some new local conservation laws for the Short Pulse equation and for the Fornberg-Whitham equation are also determined.
A class of partial differential equations (a conservation law and two balance laws), with three i... more A class of partial differential equations (a conservation law and two balance laws), with three independent variables and involving six arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent, and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, we are interested in finding an equivalent autonomous system of conservation laws. It is shown how the results apply to some physical problems.
Communications in Nonlinear Science and Numerical Simulation, May 1, 2019
In this paper, analytical and numerical solutions of time and space fractional advection-diffusio... more In this paper, analytical and numerical solutions of time and space fractional advection-diffusion-reaction equations are found. In general, by using Lie transformations, it is possible to reduce the fractional partial differential equations into fractional ordinary differential equations, if the symmetries admitted by target equations allow to determine the Lie transformations. In the case of the time and space fractional advection-diffusion-reaction model, the Lie symmetries do not lead to reduce the equation into fractional ordinary one. So we propose an alternative strategy to find the analytical and numerical solutions starting from the analytical and numerical results recently obtained by the authors for the time fractional advection-diffusion-reaction equation and for the space fractional advection-diffusion-reaction equation, separately, by using the Lie symmetries. The numerical results prove the efficiency and the applicability of the proposed procedure that results to be, for its high precision, a good tool to find solutions of a wide class of problems involving the fractional differential equations.
Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali, May 13, 2021
In this paper, exact and numerical solutions of two dimensional time-fractional diffusion-reactio... more In this paper, exact and numerical solutions of two dimensional time-fractional diffusion-reaction equation involving the Riemann-Liouville derivative are determined, by applying a procedure that combines the Lie symmetry analysis with the numerical methods. Two new reduced fractional differential equations are obtained by using the Lie symmetry theory. Applying only one Lie transformation, we get a new time-fractional partial differential equation and, applying a further Lie transformation, we get an ordinary differential equation. Numerical solutions of the reduced differential equations are computed separately by implicit numerical methods. A comparative study between numerical solutions is performed.
A class of partial differential equations (a conservation law and four balance laws), with four i... more A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered.
International Journal of Non-linear Mechanics, Oct 1, 2017
In this paper, by using a mixed approach, recently introduced by the authors, some conservation l... more In this paper, by using a mixed approach, recently introduced by the authors, some conservation laws of partial differential equations are derived. The method merges the Ibragimov's method and the one by Anco and Bluman. In particular, by applying this new mixed method, we determine all zero-th order conservation laws of Chaplygin and Shallow Water equations, as well as new conservation laws for a second order partial differential equation involving an arbitrary function.
In this paper, numerical solutions of space-fractional advection-diffusion equations, involving t... more In this paper, numerical solutions of space-fractional advection-diffusion equations, involving the Riemann-Liouville derivative with a nonlinear source term, are presented. We propose a procedure that combines the fractional Lie symmetries analysis, to reduce the original fractional partial differential equations into fractional ordinary differential equations, with a numerical method. By adopting the Caputo definition of derivative, the reduced fractional ordinary equations are solved by applying the implicit trapezoidal method. The numerical results confirm the applicability and the efficiency of the proposed approach.
INTERNATIONAL YOUTH SCIENTIFIC CONFERENCE “HEAT AND MASS TRANSFER IN THE THERMAL CONTROL SYSTEM OF TECHNICAL AND TECHNOLOGICAL ENERGY EQUIPMENT” (HMTTSC 2019), 2019
A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with... more A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with a nonlinear source term is considered. The authors determine the Lie symmetries and reduce the original fractional partial differential equation into a fractional ordinary differential equation (FODE), in some interesting cases. Numerical solutions of FODEs are obtained, and through Lie transformations, numerical solutions of original equation are found.
In this paper, by applying Valenti’s theory for the approximate symmetry, we introduce and define... more In this paper, by applying Valenti’s theory for the approximate symmetry, we introduce and define the concept of a one-dimensional optimal system of approximate subalgebras for a generalized Ames’s equation; furthermore, the algebraic structure of the approximate Lie algebra is discussed. New approximately invariant solutions to the equation are found.
In this paper, we study an integrable system of coupled KdV equations, derived by Gear and Grimsh... more In this paper, we study an integrable system of coupled KdV equations, derived by Gear and Grimshaw (Stud. Appl. Math. 70(3):235-258, 1984), modeling the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratified fluid. In particular, we present the complete group classification of the model and find conditions on arbitrary parameters for which the system admits symmetries. Some exact solutions of physical relevance are derived.
Invariant solutions of partial differential equations are found by solving a reduced system invol... more Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.
A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with... more A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with a nonlinear source term is considered. The authors determine the Lie symmetries and reduce the original fractional partial differential equation into a fractional ordinary differential equation (FODE), in some interesting cases. Numerical solutions of FODEs are obtained, and through Lie transformations, numerical solutions of original equation are found.
A new approach, combining the Ibragimov method and the one by Anco and Bluman, with the aim of al... more A new approach, combining the Ibragimov method and the one by Anco and Bluman, with the aim of algorithmically computing local conservation laws of partial differential equations, is discussed. Some examples of the application of the procedure are given. The method, of course, is able to recover all the conservation laws found by using the direct method; at the same time we can characterize which symmetry, if any, is responsible for the existence of a given conservation law. Some new local conservation laws for the Short Pulse equation and for the Fornberg-Whitham equation are also determined.
A class of partial differential equations (a conservation law and two balance laws), with three i... more A class of partial differential equations (a conservation law and two balance laws), with three independent variables and involving six arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent, and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, we are interested in finding an equivalent autonomous system of conservation laws. It is shown how the results apply to some physical problems.
Communications in Nonlinear Science and Numerical Simulation, May 1, 2019
In this paper, analytical and numerical solutions of time and space fractional advection-diffusio... more In this paper, analytical and numerical solutions of time and space fractional advection-diffusion-reaction equations are found. In general, by using Lie transformations, it is possible to reduce the fractional partial differential equations into fractional ordinary differential equations, if the symmetries admitted by target equations allow to determine the Lie transformations. In the case of the time and space fractional advection-diffusion-reaction model, the Lie symmetries do not lead to reduce the equation into fractional ordinary one. So we propose an alternative strategy to find the analytical and numerical solutions starting from the analytical and numerical results recently obtained by the authors for the time fractional advection-diffusion-reaction equation and for the space fractional advection-diffusion-reaction equation, separately, by using the Lie symmetries. The numerical results prove the efficiency and the applicability of the proposed procedure that results to be, for its high precision, a good tool to find solutions of a wide class of problems involving the fractional differential equations.
Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali, May 13, 2021
In this paper, exact and numerical solutions of two dimensional time-fractional diffusion-reactio... more In this paper, exact and numerical solutions of two dimensional time-fractional diffusion-reaction equation involving the Riemann-Liouville derivative are determined, by applying a procedure that combines the Lie symmetry analysis with the numerical methods. Two new reduced fractional differential equations are obtained by using the Lie symmetry theory. Applying only one Lie transformation, we get a new time-fractional partial differential equation and, applying a further Lie transformation, we get an ordinary differential equation. Numerical solutions of the reduced differential equations are computed separately by implicit numerical methods. A comparative study between numerical solutions is performed.
A class of partial differential equations (a conservation law and four balance laws), with four i... more A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered.
International Journal of Non-linear Mechanics, Oct 1, 2017
In this paper, by using a mixed approach, recently introduced by the authors, some conservation l... more In this paper, by using a mixed approach, recently introduced by the authors, some conservation laws of partial differential equations are derived. The method merges the Ibragimov's method and the one by Anco and Bluman. In particular, by applying this new mixed method, we determine all zero-th order conservation laws of Chaplygin and Shallow Water equations, as well as new conservation laws for a second order partial differential equation involving an arbitrary function.
In this paper, numerical solutions of space-fractional advection-diffusion equations, involving t... more In this paper, numerical solutions of space-fractional advection-diffusion equations, involving the Riemann-Liouville derivative with a nonlinear source term, are presented. We propose a procedure that combines the fractional Lie symmetries analysis, to reduce the original fractional partial differential equations into fractional ordinary differential equations, with a numerical method. By adopting the Caputo definition of derivative, the reduced fractional ordinary equations are solved by applying the implicit trapezoidal method. The numerical results confirm the applicability and the efficiency of the proposed approach.
INTERNATIONAL YOUTH SCIENTIFIC CONFERENCE “HEAT AND MASS TRANSFER IN THE THERMAL CONTROL SYSTEM OF TECHNICAL AND TECHNOLOGICAL ENERGY EQUIPMENT” (HMTTSC 2019), 2019
A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with... more A space-fractional advection-diffusion equation, involving the Riemann-Liouville derivative, with a nonlinear source term is considered. The authors determine the Lie symmetries and reduce the original fractional partial differential equation into a fractional ordinary differential equation (FODE), in some interesting cases. Numerical solutions of FODEs are obtained, and through Lie transformations, numerical solutions of original equation are found.
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