Non-Linear Partial Differential Equations: An Algebraic View of Generalized SolutionsA massive transition of interest from solving linear partial differential equations to solving nonlinear ones has taken place during the last two or three decades. The availability of better computers has often made numerical experimentations progress faster than the theoretical understanding of nonlinear partial differential equations. The three most important nonlinear phenomena observed so far both experimentally and numerically, and studied theoretically in connection with such equations have been the solitons, shock waves and turbulence or chaotical processes. In many ways, these phenomena have presented increasing difficulties in the mentioned order. In particular, the latter two phenomena necessarily lead to nonclassical or generalized solutions for nonlinear partial differential equations. |
Contents
| 1 | |
CHAPTER 2 GLOBAL VERSION OF THE CAUCHY KOVALEVSKAIA THEOREM ON ANALYTIC NONLINEAR PARTIAL DIFFERENTIAL E... | 101 |
CHAPTER 3 ALGEBRAIC CHARACTERIZATION FOR THE SOLVABILITY OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS | 131 |
CHAPTER 4 GENERALIZED SOLUTIONS OF SEMILINEAR WAVE EQUATIONS WITH ROUGH INITIAL VALUES | 173 |
CHAPTER 5 DISCONTINUOUS SHOCK WEAK AND GENERALIZED SOLUTIONS OF BASIC NONLINEAR PARTIAL DIFFERENTIAL EQUA... | 197 |
CHAPTER 6 CHAINS OF ALGEBRAS OF GENERALIZED FUNCTIONS | 221 |
CHAPTER 7 RESOLUTION OF SINGULARITIES OF WEAK SOLUTIONS FOR POLYNOMIAL NONLINEAR PARTIAL DIFFERENTIAL EQUA... | 271 |
CHAPTER 8 THE PARTICULAR CASE OF COLOMBEAUS ALGEBRAS | 301 |
Final Remarks | 367 |
| 371 | |
Other editions - View all
Non-linear Partial Differential Equations: An Algebraic View of Generalized ... Elemer E. Rosinger No preview available - 1990 |
Non-Linear Partial Differential Equations: An Algebraic View of Generalized ... E.E. Rosinger No preview available - 1990 |
Common terms and phrases
analytic nonlinear partial Appendix arbitrary assume c g(R C*-smooth functions C*-smooth regular Cauchy Cauchy-Kovalevskaia theorem chains of algebras Chapter classical solution cº(R coefficients Colombeau 1,2 construction continuous functions convergent derivative operators differential algebra e g(R embedding equivalence relation existence fact follows easily framework Further global hence implies inclusion diagrams initial value problem instance large class Lemma let us define let us denote linear partial differential mapping mentioned method nonlinear partial differential nonlinear theory nonvoid numbers Oberguggenberger obtain obviously partial derivatives partial differential equations partial differential operator particular polynomial nonlinear partial proof Proposition quotient algebras resolution of singularities Rosinger satisfies Schwartz distributions Section smooth functions stability paradoxes structure subalgebra subset Suppose given tions topology unique usual vector space vector subspaces weak solution x e Q yields