Evolution Equations with a Complex Spatial Variable

The following sections are included:

• Dedication
• Preface
• Contents

In this introductory chapter we present the main historical background and motivations supporting the writing of this book.

In this chapter we study the one-dimensional heat and Laplace equations with real temporal variable and complex spatial variable. All the results presented in this chapter are contained in Gal-Gal-Goldstein [32], except for Theorems 2.2.8 and 2.2.9 which are new.

The representation in Theorem 2.2.1, (i), of Chapter 2, suggests attaching to

In Chapter 2, the classical heat and Laplace equations with real time variable and complex spatial variable were studied. The purpose of the present chapter is to make an analogous study for wave and telegraph equations with a real time variable and a complex spatial variable. The complexification of the spatial variable in the wave and telegraph equations is made by the same two methods which produce different equations. By the first method, we complexify the spatial variable in the corresponding formulas by replacing the usual translations x ± ct, c is the speed of propagation, by the rotations ze^±ict and, by the second method, we complexify the spatial variable in the corresponding evolution equation and then we search for analytic and non-analytic solutions. The first method produces solutions that also preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. Moreover, new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated from both methods and their solutions are constructed. All the results contained in this chapter are taken from Gal-Gal-Goldstein [34].

The purpose of this chapter is to study the Burgers equation and Black-Merton-Scholes equation with real time variable and complex spatial variable. As in the previous chapters, the complexification of the spatial variable in these equations is made by two different methods which produce different equations. First, one complexifies the spatial variable in the corresponding (real) solution by replacing the usual sum of variables (translation) by an exponential product (rotation) and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions…

The purpose of this chapter is to study the classical Schrödinger equation with real time variable and complex spatial variable. The complexification of the spatial variable is made by the two different methods which were also employed in the previous chapters. All the results contained in this chapter are taken from Gal-Gal-Goldstein [36].

The purpose of this chapter is to apply both methods of “complexification” to the linearized Korteweg-de Vries equation with a real time variable and a complex spatial variable. All the results in this chapter are contained in [37].

In this chapter we consider the evolution equations from the previous chapters on general domains Ω of the complex plane. Some examples include simply connected domains bounded by a regular analytic curve, or even continua (that is, compact connected subsets). All the results in this chapter, except where otherwise mentioned, appear here for the first time.

The following sections are included:

• Bibliography
• Index