Differential Equations:       An Operational Approach
Héctor Manuel Moya-Cessa Francisco Soto-Eguibar
164 pages, 9x6 inches
June 2011 Paperback
ISBN 978-1-58949-060-4



In this short textbook on differential equations an alternative approach to the one that is usually found in textbooks is presented. Original material that deals with the application of operational methods is developed. Particular attention is paid to algebraic methods by using differential operators when applicable. The methods presented in this book are useful in all applications of differential equations. These methods are used in quantum physics, where they have been developed in their majority, but can be used in any other branch of physics and engineering.

undergraduate students, graduate students, teachers, researchers in modern and applied physics and engineering.



 1  Linear Algebra
1.1 Vector spaces
1.1.1 Subspaces
1.1.2 The span of a set of vectors
1.1.3 Linear independence
1.1.4 Bases and dimension of a vector space

1.1.5 Coordinate systems. components of a vector in a given basis
1.2 The scalar product. Euclidian spaces

1.2.1 The norm in an Euclidian space
1.2.2 The concept of angle between vectors. Orthogonality
1.3 Linear transformations

1.3.1 The kernel of a linear transformation
1.3.2 The image of a linear transformation
1.3.3 Isomorphisms
1.3.4 Linear transformations and matrices

1.3.5 The product of linear transformations

1.4. Eigenvalues and eigenvectors
1.4.1 The finite dimension case
1.4.2 Similar matrices
1.4.3 Diagonal matrices Procedure to diagonalize a matrix The Cayley-Hamilton thorem

1.5. Linear operators acting on Euclidian spaces
1.5.1 Adjoint operators

1.5.2 Hermitian and anti-Hermitian operators
1.5.3 Properties of the eigenvalues and eigenvectors of the Hermitian operators

2  Special Functions
2.1 Hermite polynomials
.1.1 Baker-Hausdorff formula
2.1.2 Series of even Hermite polynomials
2.1.3 Addition formula

2.2 Associated Laguerre polynomials
2.3 Chebyshev polynomials
2.3.1 Chebyshev polynomials of the first kind
2.3.2 Chebyshev polynomials of the second kind
2.4 Bessel functions of the first kind of integer order
2.4.1 Addition formula
2.4.2 Series of teh Bessel functions of the first kind of integer order 
2.4.3 Relation between the Bessel functions of the first kind of integer order and the Chebyshev polynomials of the second kind

Finite Systems of Differential Equations
3.1 Systesms 2x2 first
3.1.1 Eigenvalue equations
3.1.2 Cayley-Hamilton theorem method case one case two
3.2 Systems 4x4
3.2.1 Case one
3.2.2 Case two
3.2.3 Case three
3.2.4 Case four
3.3 Systems nxn
3.3.1 All the eigenvalues are distinct
3.3.2 other cases

4  Infinite Systems of Differential Equations
4.1 Addition formula for the Bessel functions
4.2 First neighbors interaction
4.3 Second neighbors interaction
4.4 First neighbor interaction with an extra interaction
4.4.1 Interaction omega-n
4.4.2 Interaction omega-(-1)^n

5  Semi-infinite Systems of Differential Equations
5.1 First a semi-infinite system
5.2 Semi-infinite system with first neighbors interaction
5.3 Nonlinear system with first neighbors interaction

6  Partial Differential Equations
6.1 A simple partial differential equation
6.1.1 A Gaussian function as boundary condition
6.1.2 An arbitrary function as boundary condition
6.2 Airy system
6.2.1 Airy function as boundary condition
6.3 Harmonic oscillator system
6.4 z-dependent harmonic oscillator

Appendix A  Dirac Notation

Appendix B  Inverse of the Vandermonde and Vandermonde Confluent Matrices
B.1 The inverse of the Vandermonde matrix
B.2 The inverse of the confluent Vadermonde matrix  

Appendix C  Tridiagonal matrices
C.1 Fibonacci system