This monograph is devoted to new types of higher order PDEs in the framework of Clifford analysis. While elliptic and hyperbolic equations have been studied in the Clifford analysis setting in book and journal literature, parabolic equations in this framework have been largely ignored and are the primary focus of this work. Thus, new types of equations are examined: elliptic-hyperbolic, elliptic-parabolic, hyperbolic-parabolic and elliptic-hyperbolic-parabolic. These equations are related to polyharmonic, polywave, polyheat, harmonic-wave, harmonic-heat, wave-heat and harmonic-wave-heat equations for which various boundary and initial value problems are solved explicitly in quadratures. The solutions to these new equations in the Clifford setting have some remarkable applications, for example, to the mechanics of deformable bodies, electromagnetic fields, and quantum mechanics.
Introduction.- Part I: Boundary Value Problems for Regular, Generalized, Regular and Pluriregular Elliptic Equations.- Two Dimensional Cases; Multi-Dimensional Cases.- Part II: Initial Value Problems for Regular, Pluriregular Hyperbolic and Parabolic Equations.- Hyperbolic and Plurihyperbolic Equations in Clifford Analysis; Parabolic and Pluriparabolic Equations in Clifford Analysis; Epilogue.- References.- Index.