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Роботын байршлыг хурдтайгаар, бодит хугацаанд олоход ротари энкодер мэдрүүлийг ашиглан нь үр дүнтэй байдаг. Энэ судалгааны ажил дээр хоёр болон гурван чөлөөний зэрэг бүхий роботыг байршлыг ротари энкодер ашиглан олох аргыг өгнө.
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. A group of 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 their particular cases.
We derive explicit solutions for time-fractional anomalous diffusion equations with diffusivity coefficients that depend on both space and time variables. These solutions are expressed in Fox-H and generalized Wright functions, which are commonly used in anomalous diffusion equations. Our study represents a significant advancement in our understanding of anomalous diffusion with potential applications in a wide range of fields.
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.
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.
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 article, we study the group analysis of time fractional generalized Fisher equations. The generalized equation includes nonlinear heat and wave equations with a source as particular cases. A complete group classification is obtained and we also constructed group invariant solutions corresponding to infinitesimal symmetries. Explicit solutions had been found for some cases.
There is a recurrence relation for calculating an integral of associated Legendre function Pml. Here we try to find a direct formula for the integral by using generating function and we successfully developed a formula for m = l case.
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.
We study time fractional linear and nonlinear evolution systems with variable coefficients via Lie symmetry analysis. To study the fractional systems, we have obtained formulas for extended infinitesimals of a system of fractional differential equations. For linear systems, we not only give complete group classification of invariant solutions but also exact solutions corresponding to infinitesimal symmetries of optimal systems of Lie algebras generated by infinitesimal symmetries. The group invariant solutions are expressed in terms of three kinds of special functions: the Mittag-Leffler functions, the generalized Wright functions and the Fox H functions. Which of these special functions, we use in any given case depends on the order of the fractional derivative and right-hand sides of the equations in the fractional linear system. For fractional nonlinear evolution system, we give complete group classification along with explicit invariant solutions in some particular cases.
During last years differential equations with derivatives of fractional order have gained increasing popularity. Such equations and their systems accurately model various nonlinear phenomena in many fields including wave studies, diffusion processes and fluid mechanics, etc. Hence, a number of effective techniques have been developed to construct exact solutions of these equations such as G-expansion type methods, variational iteration methods, Adomian decomposition methods, Lie symmetry methods and exp-function methods, among others. In particular, Lie symmetry analysis provides a generic and efficient algorithmic approach for corresponding exact solutions to fractional partial differential equations (FPDEs) and systems of FPDEs in explicit forms. The key idea of the Lie symmetry analysis is regarding the tangent structural equations under one or several parameters Lie groups of point transformations. One can construct exact solutions including similarity solutions or more general group-invariant solutions by corresponding symmetry reductions.
Fast and efficient integer factorization algorithms are crucial in analyzing, designing and breaking cryptographic schemes. The well-known quadratic sieve algorithm utilizes nontrivial solutions of the equation x2 − y2 = 0 in Zn. It uses non-deterministic method which called seiving to construct the non-trivial solutions. Naturally, the running time of the quadratic sieve algorithm for a given integer n depends on the ratio of the cardinality of non-trivial solutions and trivial solutions, i.e. x = ±y in Zn. In this paper, we propose an efficient algorithm for counting the non-trivial as well as trivial solutions. It is believed that the number of non-trivial solutions are at least half of the solutions satisfying congruent of squares. But we’ll show that it’s not the case for the numbers in RSA cryptosystem in which the product of two distinct primes are used, by calculating the exact number of trivial and non-trivial solutions using our developed algorithm. If the number of distinct primes are increased, the number of non-trivial solutions are more than the number of trivial solutions.
A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi in 1996. The investigated system of partial differential equations generalizes the Gauss-Aomoto-Gelfand system. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of 2 2 matrices of derivations with respect to certain variables. The Gauss- Aomoto-Gelfand hypergeometric system arises in numerous problems of algebraic geometry, partial differential equations, the theory of special functions and combinatorics. H. Sekiguchi generalized this construction by looking at determinants of ll matrices of derivations with respect to certain variables. In this poster we studied the dimension of global solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems.
файлаар хавсаргав
файлаар хавсаргав.
файлаар оруулав
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.
файлаар оруулав
https://smns.msue.edu.mn/Category/Post/86ad91a8-1229-4fcd-9ea6-0dde929e8a0d
A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi in 1996. The investigated system of partial differential equations generalizes the Gauss-Aomoto-Gelfand system. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of $2\times 2$ matrices of derivations with respect to certain variables. The Gauss-Aomoto-Gelfand hypergeometric system arises in numerous problems of algebraic geometry, partial differential equations, the theory of special functions and combinatorics. H. Sekiguchi generalized this construction by looking at determinants of $l\times l$ matrices of derivations with respect to certain variables. In this talk we studied the dimension of global solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems.
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.
We give conservation laws for time fractional nonlinear telegraph equations.
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.
Exact solutions of time fractional linear diffusion-wave equations
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.
