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We study a class of time fractional diffusion-wave equations with variable coefficients using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of equations. Group invariant solutions are given explicitly corresponding to every element in an optimal system of Lie algebras generated by infinitesimal symmetries of equations in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions. These solutions contain previously known solutions as particular cases.
In this study, we present discretization schemes based on Generalized Integral Representation Method (GIRM) for numerical simulation of the Boussinesq wave. The schemes numerically evaluate the coupled Boussinesq equation for different solitary wave phenomena, namely, propagation of a single soliton, head-on collision of two solitons and reflection of a soliton at a fixed wall boundary. In these three soliton interactions, we utilize different Generalized Fundamental Solutions (GFS) along with piecewise constant approximations for the unknown functions. For the case of soliton reflection at a wall, time evolution in GIRM is coupled with the Green’s function in order to cope with the complicated boundary conditions that arise from the GIRM derivation. We conduct numerical experiments and obtain satisfactory approximate result for each case of the soliton interactions.
In order to more accurately describe the dynamics of complex processes with long-term memory, spatial heterogeneity, as well as nonstationary and nonergodic statistics, time fractional differential equations are often used. We take into consideration fractional differential equations that explain anomalous diffusivity and are of the diffusion-wave type with positive real order of time derivative. Anomalous diffusion can now be used to explain a growing number of processes in nature, as well as in technological and social science. In this talk, we present our results on exact, analytical answers to fractional diffusion-wave equations that are both linear and nonlinear with various variable coefficients. The answers are explicitly expressed in Fox H and generalized Wright functions and are invariant under certain transformations. We also demonstrate how, for particular parameter values, we can derive some well known solutions to the heat and wave equations.
Here we present a simple numerical scheme for simulation of the coupled Boussinesq equation. The scheme is based on Generalized Integral Representation Method and numerically simulate the Boussinesq wave for reflection of a solitary wave at a vertical wall fixed on the boundary. In this particular wave interaction, we utilize two different Generalized Fundamental Solutions (GFS) in our scheme, namely, common Gaussian GFS and harmonic GFS, for comparison purpose. The unknown functions in the Boussinesq equation are expressed via piecewise constant approximations. We emphasize that time evolution in the scheme, is coupled with the Green’s function in order to cope with the complicated boundary conditions that arise during derivation of the numerical scheme.
Anomalous diffusion has been the subject of extensive research in recent years with numerous oublications addressing different aspects of this phenomenon. We provided exact solutions for anomalous diffusion equations with a diffusion coefficient function. We derive closed form solutions for time fractional anomalous diffusion equations with diffusivity coefficients that depend on both space and time variables
In this work we studied a generalized Fisher equation in cylindrical coordinate using Lie symmetry method. We have determined for what type of source function the generalized Fisher equation has Lie Symmetries other than time translation symmetry when the diffusion function is given by an exponential function. Also the reduced ordinary differential equations are obtained corresponding to Lie symmetries of the generalized Fisher equation.
The Boussinesq wave equation is a simpli ed model used to describe the propagation of long waves in shallow water. This equation combines the effects of dispersion and nonlinear wave interactions. Solving the Boussinesq equation can be challenging, and analytical solutions are often not feasible. Various numerical methods, such as nite di erence, nite element, or spectral methods, are commonly employed to approximate the solution. On the other hand, Generalized Integral Representation Method (GIRM) is a mathematical technique used for solving differential equations, integral equations, and partial di erential equations. The GIRM technique involves expressing the solution of a given problem as a weighted integral of a known function. This integral representation can then be used to obtain the solution for any point within the domain of the problem. In this study, we discuss our discretization schemes based on GIRM for numerical study of the 1D Boussinesq wave in case of soliton reflection at a fixed boundary.
We study a certain class of time-fractional diffusion-wave equations with variable coefficients via Lie symmetry analysis. In physics, fractional diffusion equation describes transport dynamics that are governed by anomalous diffusion while fractional wave equation describes oscillations and wave propagation in various physical systems. In order to obtain exact invariant solutions of these equations, we firstly determine infinitesimal symmetries in respect to the variable coefficients of the equations. With the help of these symmetries, we then find new solutions in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions.
In this work, we use Newton-like methods to solve systems of nonlinear equations numerically. This work will be divided in 2 parts: Theoretical and Application part. The theoretical part covers the study of convergence of two and three-step Newton-like methods. Let us find the root x of F(x) = 0 here F is a system of nonlinear equations.
We discuss of computer implementation of Generalized Integral Representation Method (GIRM) for one-dimensional diffusion problem on regular meshes. Although GIRM requires the initial matrix inversion of the given problem, the solution is stable and the accuracy is satisfactory. Moreover, it can be applied to an irregular mesh. In order to confirm the theory, we apply GIRM to the one-dimensional Initial and Boundary Value Problem for advective diffusion equation. The numerical experiments are conducted and the approximate solutions coincide with the exact ones in both cases. The computer code is implemented in Matlab is given and discussed in detail.
We present discretization schemes based on Generalized Integral Representation Method (GIRM) for numerical study of the Boussinesq wave. The schemes numerically evaluate the coupled Boussinesq equation for three different solitary wave phenomena, namely, propagation of a single soliton, head-on collision of two solitons and reflection of soliton at a fixed boundary. In each case of the soliton interactions, we utilize different Generalized Fundamental Solutions (GFS) along with piecewise constant approximations for the unknown functions.
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In this talk, we study a class of linear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems.
https://smns.msue.edu.mn/Category/Post/86ad91a8-1229-4fcd-9ea6-0dde929e8a0d
We study a class of time fractional diffusion-wave equations with variable coefficients using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of equations. Group invariant solutions are given explicitly corresponding to every element in an optimal system of Lie algebras generated by infinitesimal symmetries of equations in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions. These solutions contain previously known solutions for particular cases.
Active contour models with fractional order derivative has studied in last few years due to the object detection becoming more accurate than integer order derivative. In this work, we consider the energy functional consists of two terms : fitting term and regularization term. Fractional order fitting term and global fitting term can describe the original image more accurately. We use Grunwald-Letnikov fractional order differentiation. Moreover, we present the two-dimensional fractional order differentiation, that was acquired through the extension of the one-dimensional fractional order differentiation. Finally, we present some experimental results which are compared to base model with integer order derivative.
In this study, we consider a nonlinear telegraph system of time--fractional equations by using Lie symmetry analysis. Fractional derivative defined by the Riemann--Liouville operator is considered for two cases, in each of which symmetry generators are found. Through these generators group--invariant solutions are given in explicit forms. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. Finally, conservation laws for the system are extracted.
It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling transformations. We then derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusionwave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases.
Хураангуй: We explicitly give new group invariant solutions to a class of Riemann-Liouville time fractional evolution systems with variable coefficients. These solutions are derived from every element in an optimal system of Lie algebras generated by infinitesimal symmetries of evolution systems in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions and show that these solutions solve diffusion-wave equations with variable coefficients. These solutions contain previously known solutions as particular cases. Some plots of solutions subject to the order of the fractional derivative are illustrated.
In this study, we consider a nonlinear telegraph system of time--fractional equations by using Lie symmetry analysis. Fractional derivative defined by the Riemann--Liouville operator is considered for two cases, in each of which symmetry generators are found. Through these generators group--invariant solutions are given in explicit forms.
We study a class of time fractional diffusion-wave equations with variable coefficient using Lie symmetry analysis. We obtain a complete group classification and a classification of group invariant solutions of this class of equations. The reduced equations corresponding to the optimal systems of Lie algebras of infinitesimal symmetries are also obtained. Group invariant solutions are found explicitly using our complementary work to this work. The invariant solutions are expressed in means of special functions
We derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of generalizedWright functions and Fox H-functions. These solutions are invariant solutions of diffusionwave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases.
