A Hodograph Transformation Applicable to a Large Class of PDEs

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1986

A method of performing hodograph transformations of partial derivatives of any order lor arbitrary dimensions is worked out. Hodograph transformation means a change of the roles of arguments and functions. The formulas obtained for first derivatives can easily be applied to any dimensions. Several explicit formulas for lower orders and dimensions are also given. Application to partial differential equations is discussed.

Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics, 1979

A phenomenon present in many physical problems hodograph invariance of partial differential equations is considered for two-dimensional submanifolds in three-and four-dimensional spaces. This means that the equation keeps its shape when interchanging the roles of any of the functions and arguments, i. e. under a hodograph transformation. For that purpose unified formulations for hodograph transformations are obtained. A method for the formation of hodograph invariant equations is shown. The well-known scalar Born-Infeld equation and its two-component generalization prove

Physics Letters A Volume 320, Issues 5–6, 12 January 2004, Pages 383–388

A hodograph transformation for a wide family of multidimensional nonlinear partial differential equations is presented. It is used to derive solutions of the Boyer–Finley equation (dispersionless Toda equation), which are not group invariant, and the corresponding family of explicit ultra-hyperbolic selfdual vacuum spaces.

2002

A hodograph transformation for a wide family of multidimensional nonlinear partial differential equations is presented. It is used to derive solutions of the heavenly equation (dispersionless Toda equation) as well as a family of explicit ultra-hyperbolic selfdual vacuum spaces admiting only one Killing vector which is not selfdual, we also give the corresponding explicit Einstein--Weyl structures.

Journal of King Saud University - Science, 2019

In this work, a new algorithm is proposed for computing the differential transform of two-dimensional nonlinear functions. This algorithm overcomes the drawbacks of previous algorithms as it is straightforward for any form of analytic nonlinearities and does not require any intermediate calculations or algebraic manipulations. This is accomplished by defining a new form for two dimensional polynomials that generalize the differential transform of the corresponding one-dimensional function to higher dimensions. The correctness of this algorithm is proved via the multivariable Faa di Bruno formula. Several examples with different types of nonlinearities are solved to verify the efficiency of the proposed algorithm.

Journal of Mathematical Analysis and Applications, 2001

We introduce nonlocal auto-hodograph transformations for a hierarchy of nonlinear evolution equations. This is accomplished by composing nonlocal transfor-Ž . mations one of which is a hodograph transformation which linearize the given equations. This enables one to construct sequences of exact solutions for any equation belonging to the hierarchy.

CHAPTER 3. FURTHER NEW RESULTS ON N-DEMENSIONAL LAPLACE AND INVERSE LAPLACE TRANSFORMATIONS 3.1. Introduction 3.2. The Image of Functions with the Argument ' 89 3.2.1. Applications of Theorem 3.2.1 3.2.2. Laplace Transforms of some Elementary and Special Functions with n Variables ^ 98 3.3. The Original of Functions with the Argument ^ 106 3.3.1. Examples Based Upon Theorem 3.3.1 Ill 3.4. The Image of Functions with the Argument 2pi(x 114 3.4.1 Applications of Theorem 3.4.1 119 3.5. The Original of Functions with the Argument 121 3.5.1. Example Based Upon Theorem 3.5.1 125 CHAPTER 4. THE SOLUTION OF INITIAL-BOUNDARY-VALUE PROBLEMS (IBVP'S) BY DOUBLE LAPLACE TRANSFORMATIONS 127 4.1. Introduction 127 4.2. Non-homogenous Linear Partial Differential Equations (PDEs) of the First Order 129 4.2.1. Partial Differential Equations of Type Ux+u.y = f{x,y),Q<x<«o,0<y<oo 129 4.2.2. Partial Differential Equations of Type au^ + buy + eeu = fix,y), 0<a:<<», 0<y<~ 136 4.3. Non-homogenous Second Order Linear Partial Differential V Equations of Hyperbolic Type 4.3.1. Partial Differential Equations of Type Ujcy = f(x,y),),0<x«>o,0<y<oo 4.3.2. The Wave Equation 4.4. Non-homogenous Second Order Partial Differential Equations of Parabolic Type 4.4.1. Partial Differential Equations of Type u"+2u^+Uyy + KU = fix,y\0<x<oo,0<y<oo CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS 154 5.1. Conclusions 154 5.2. Future Directions 155

Indian Journal of Science and Technology

In this work, an analytical solution of linear and nonlinear multidimensional partial differential equation is deduced by the Differential Transform Method (DTM). Some numerical examples are presented to demonstrate the efficiency and reliability of this method.

Physics Letters A, 2008

Applied Mathematics and Computation, 2001

The dierential transform is a numerical method for solving dierential equations. In this paper, we present the de®nition and operation of the two-dimensional dierential transform. A distinctive feature of the dierential transform is its ability to solve linear and nonlinear dierential equations. Partial dierential equation of parabolic, hyperbolic, elliptic and nonlinear types can be solved by the dierential transform. We demonstrate that the dierential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial dierential equation.