Equations equivalent to the classical one-dimensional heat equation by reciprocal transformations
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Abstract
This paper investigates the equivalence of parabolic partial differential equations to the classical one-dimensional heat equation using reciprocal transformations. The equations are assumed to be autonomous, and the methodology applied is similar to S. Lie’s approach to solving the linearization problem of second-order ordinary differential equations. The research is structured in two main parts. In the first part, necessary constraints on the class of parabolic partial differential equations with two independent variables, which are equivalent to the classical heat equation under a reciprocal transformation, are identified. In the second part, the remaining conditions are examined, and sufficient conditions are derived. The corresponding differential equations are then obtained. All possible cases that arise are thoroughly analyzed, and the theory is illustrated with several examples.
Abstract
This paper investigates the equivalence of parabolic partial differential equations to the classical one-dimensional heat equation using reciprocal transformations. The equations are assumed to be autonomous, and the methodology applied is similar to S. Lie’s approach to solving the linearization problem of second-order ordinary differential equations. The research is structured in two main parts. In the first part, necessary constraints on the class of parabolic partial differential equations with two independent variables, which are equivalent to the classical heat equation under a reciprocal transformation, are identified. In the second part, the remaining conditions are examined, and sufficient conditions are derived. The corresponding differential equations are then obtained. All possible cases that arise are thoroughly analyzed, and the theory is illustrated with several examples.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- About the editors
- List of contributors XI
- Vortex structures in a two-dimensional isotropic ferromagnet 1
- On the wave propagation without reflection in inhomogeneous medium: traveling Alfvén waves in plasma flows along magnetic field 17
- Exponential map of germs of vector fields 37
- The integral representation of Binet type 43
- Model and bending analysis of the sandwich beam with composite facings and compressible orthotropic core using Abramov sweep method 53
- Sedeonic equations for hydrodynamic model of electron fluid in superconductors 79
- Modified elliptic equation for heat transfer in solids 101
- Approximate solution for the far fields of momentumless turbulent wakes 117
- Equations equivalent to the classical one-dimensional heat equation by reciprocal transformations 129
- On nonlinear resonance in vector derivative nonlinear Schrödinger equations 151
- On the lifetime of free neutrons and transformations of time intervals in inertial reference systems 163
- Index 169
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- About the editors
- List of contributors XI
- Vortex structures in a two-dimensional isotropic ferromagnet 1
- On the wave propagation without reflection in inhomogeneous medium: traveling Alfvén waves in plasma flows along magnetic field 17
- Exponential map of germs of vector fields 37
- The integral representation of Binet type 43
- Model and bending analysis of the sandwich beam with composite facings and compressible orthotropic core using Abramov sweep method 53
- Sedeonic equations for hydrodynamic model of electron fluid in superconductors 79
- Modified elliptic equation for heat transfer in solids 101
- Approximate solution for the far fields of momentumless turbulent wakes 117
- Equations equivalent to the classical one-dimensional heat equation by reciprocal transformations 129
- On nonlinear resonance in vector derivative nonlinear Schrödinger equations 151
- On the lifetime of free neutrons and transformations of time intervals in inertial reference systems 163
- Index 169
