Symmetry Analysis of Differential Equations: An Introduction
Daniel J. Arrigo
Format: PDF / Kindle (mobi) / ePub
A self-contained introduction to the methods and techniques of symmetry analysis used to solve ODEs and PDEs
Symmetry Analysis of Differential Equations: An Introduction presents an accessible approach to the uses of symmetry methods in solving both ordinary differential equations (ODEs) and partial differential equations (PDEs). Providing comprehensive coverage, the book fills a gap in the literature by discussing elementary symmetry concepts and invariance, including methods for reducing the complexity of ODEs and PDEs in an effort to solve the associated problems.
Thoroughly class-tested, the author presents classical methods in a systematic, logical, and well-balanced manner. As the book progresses, the chapters graduate from elementary symmetries and the invariance of algebraic equations, to ODEs and PDEs, followed by coverage of the nonclassical method and compatibility. Symmetry Analysis of Differential Equations: An Introduction also features:
- Detailed, step-by-step examples to guide readers through the methods of symmetry analysis
- End-of-chapter exercises, varying from elementary to advanced, with select solutions to aid in the calculation of the presented algorithmic methods
Symmetry Analysis of Differential Equations: An Introduction is an ideal textbook for upper-undergraduate and graduate-level courses in symmetry methods and applied mathematics. The book is also a useful reference for professionals in science, physics, and engineering, as well as anyone wishing to learn about the use of symmetry methods in solving differential equations.
3.8 gives rise to Setting the term and simplifying gives1 whose general solution is where is an arbitrary constant. Passing through the transformation 3.24 gives rise to the following exact solutions to 3.8 By choosing different constants in the infinitesimals 3.16 , we undoubtedly will obtain new transformed PDEs, from which we would obtain ODEs to solve which could lead to new exact solutions to the original equation. A natural question is: can we bypass the stage of introducing the new
symmetry reduction of the original PDE. We will consider two examples. Example 3.17a Symmetry Reduction 1 If we set in 3.89 where is an arbitrary constant, then we obtain the invariant surface condition By the method of characteristics, we obtain the solution as Substitution into the original equation 3.85 gives rise to the ODE where the argument is defined as . Integrating once and suppressing the constant of integration gives 3.90 One particular solution of 3.90 is where the constant
the original equation 3.138 gives noting that we have reduced the number of independent variables. Example 3.21b Reduction 2 . The remaining In this case, the invariant surface condition 3.137 becomes whose solution is Substitution in the original equation 3.138 gives where and . Example 3.21c Reduction 3 In this case, the invariant surface condition 3.137 becomes whose solution is Substitution in the original equation 3.138 gives where . Example 3.22 Here we calculate the
a potential function such that 3.147 shows that 3.146c becomes 3.148 From 3.146d , we obtain directly giving 3.149 Substitution of 3.147 and 3.149 into the remaining determining equations 3.146a noting 3.148 gives which easily integrates giving Thus, the infinitesimals for 3.144 are 3.150 where satisfies 3.148 . Example 3.22a Reduction 1 If we choose and in 3.150 , then the invariant surface condition is The solution of this is and substitution into the original equation 3.144
additional constant that should not have been there. Inserting this result into 2.159b yields Grouping like terms in powers of gives As this equation must be satisfied for all values of andthe fact that is independent of , then which leads to and , noting that 2.162a is automatically satisfied. Therefore, the infinitesimals are in the given equation: 2.163 Of course, other infinitesimals could be found. In fact, there is an infinite set of infinitesimals.Our next task is to find a change of