Second-order fourth-degree Painlevé-type equations

Transformations that involve a Fuchsian-type equation are used to obtain one-to-one correspondence between the Painlevé I-IV equations and certain second-order fourth-degree Painlevé-type equations.Transformations that involve a Fuchsian-type equation are used to obtain one-to-one correspondence between the Painlevé I-IV equations and certain second-order fourth-degree Painlevé-type equations

Schlesinger transformations for the second members of PII and PIV hierarchies

In this paper, we give a method to obtain the Schlesinger transformations for the second members of second and fourth Painlev´e hierarchies. The procedure involves formulating a Riemann–Hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same.In this paper, we give a method to obtain the Schlesinger transformations for the second members of second and fourth Painlev´e hierarchies. The procedure involves formulating a Riemann–Hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

On special solutions of second and fourth Painlevé hierarchies

In this article, we give special solutions of second and fourth Painlev´e hierarchies derived by Gordoa, Joshi, and Pickering. We show that for certain choice of the parameters each n-th member of these hierarchies has a special solution in terms of an n-th order differential equation. Furthermore we derive a relation between these two hierarchies.In this article, we give special solutions of second and fourth Painlev´e hierarchies derived by Gordoa, Joshi, and Pickering. We show that for certain choice of the parameters each n-th member of these hierarchies has a special solution in terms of an n-th order differential equation. Furthermore we derive a relation between these two hierarchies

Backlund transformations for discrete Painleve equations: Discrete P-II-P-V

Cataloged from PDF version of article.Transformation properties of discrete Painleve´ equations are investigated by using an algorithmic method. This method yields explicit transformations which relates the solutions of discrete Painleve´ equations, discrete PII–PV, with different values of parameters. The particular solutions which are expressible in terms of the discrete analogue of the classical special functions of discrete Painleve´ equations can also be obtained from these transformations. 2005 Elsevier Ltd. All rights reserved

Schlessinger Transformations for Painleve VI equation

Cataloged from PDF version of article.A method to obtain the Schlesinger transformations for Painlevi VI equation is given. The procedure involves formulating a Riemann-Hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same. 0 1995 American Institute of Physics

Second-order second-degree Painleve equations related with Painleve I-IV equations and Fuchsian-type transformations

Cataloged from PDF version of article.One-to-one correspondence between the Painlevé I-VI equations and certain second-order second-degree equations of Painlevé type is investigated. The transformation between the Painlevé equations and second-order second-degree equations is the one involving the Fuchsian-type equation. © 1999 American Institute of Physics

SYMMETRIES AND EXACT SOLUTIONS OF CONFORMABLE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper Lie group analysis is used to investigate invariance properties of nonlinear fractional partial differential equations with conformable fractional time derivative. The analysis is applied to Korteweg-de Vries, modified Korteweg-de Vries, Burgers, and modified Burgers equations. For each equation, all of the vector fields and the Lie symmetries are obtained. Moreover, exact solutions are given to these equations.In this paper Lie group analysis is used to investigate invariance properties of nonlinear fractional partial differential equations with conformable fractional time derivative. The analysis is applied to Korteweg-de Vries, modified Korteweg-de Vries, Burgers, and modified Burgers equations. For each equation, all of the vector fields and the Lie symmetries are obtained. Moreover, exact solutions are given to these equations

B¨acklund transformations for Cosgrove’s equation F-XVIII

In this paper we study B¨acklund transformations (BTs) for Cosgrove’s equation F-XVIII.We use the generalization of Fokas and Ablowitz method to derive BTs between F-XVIII and new fourth-order ordinary differential equations (ODEs) of Painlev´e-type. Moreover we derive auto-BT and give special solutions for F-XVIII.In this paper we study B¨acklund transformations (BTs) for Cosgrove’s equation F-XVIII.We use the generalization of Fokas and Ablowitz method to derive BTs between F-XVIII and new fourth-order ordinary differential equations (ODEs) of Painlev´e-type. Moreover we derive auto-BT and give special solutions for F-XVII

ON TAYLOR DIFFERENTIAL TRANSFORM METHOD FOR THE FIRST PAINLEVE EQUATION

We apply the Taylor Differential Transform Method (TDTM) to the initial value problem of the fi rst Painleve equation. We use the deviation to calculate the accuracy of the solutions and the results are compared with the known results. Four sets of initial values, two of them were not considered before, are considered to illustrate the effectiveness of the method.We apply the Taylor Differential Transform Method (TDTM) to the initial value problem of the fi rst Painleve equation. We use the deviation to calculate the accuracy of the solutions and the results are compared with the known results. Four sets of initial values, two of them were not considered before, are considered to illustrate the effectiveness of the method

First-order second-degree equations related with Painlevé equations

The first-order second-degree equations satisfying the Fuchs theorem concerning the absence of movable critical points, related with Painlev´e equations, and one-parameter families of solutions which solve the first-order second-degree equations are investigated.The first-order second-degree equations satisfying the Fuchs theorem concerning the absence of movable critical points, related with Painlev´e equations, and one-parameter families of solutions which solve the first-order second-degree equations are investigated