Nonlinear Differential Equations
Published on 3. December 2014
360 pages
978-1-4832-7837-7 (ISBN)
Description
Studies in Applied Mathematics, 2: Nonlinear Differential Equations focuses on modern methods of solutions to boundary value problems in linear partial differential equations. The book first tackles linear and nonlinear equations, free boundary problem, second order equations, higher order equations, boundary conditions, and spaces of continuous functions. The text then examines the weak solution of a boundary value problem and variational and topological methods. Discussions focus on general boundary conditions for second order ordinary differential equations, minimal surfaces, existence theorems, potentials of boundary value problems, second derivative of a functional, convex functionals, Lagrange conditions, differential operators, Sobolev spaces, and boundary value problems. The manuscript examines noncoercive problems and vibrational inequalities. Topics include existence theorems, formulation of the problem, vanishing nonlinearities, jumping nonlinearities with finite jumps, rapid nonlinearities, and periodic problems. The text is highly recommended for mathematicians and engineers interested in nonlinear differential equations.
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Content
PrefaceList of SymbolsChapter I. Some Examples to Begin with Section 1. Various Notations. Linear Equations Section 2. Nonlinear Equations Section 3. Nonlinear Systems Section 4. Further Nonlinear Problems Section 5. A Free Boundary Problem. The Plate EquationChapter II. Introduction Section 6. Second Order Equations Section 7. Higher Order Equations Section 8. Spaces of Continuous Functions. Solution of a Differential Equation Section 9. Boundary Conditions Section 10. Solution of a Boundary Value Problem Section 11. On an Integral IdentityChapter III. The Weak Solution of a Boundary Value Problem Section 12. The Carathéodory Property and the Nemyckii Operators Section 13. Sobolev Spaces Section 14. Differential Operators Section 15. Boundary Value Problems Section 16. Various Generalizations Section 17. Regularity of the Weak SolutionChapter IV. The Variational Method Section 18. First Derivative of a Functional Section 19. Potentials of Boundary Value Problems Section 20. The Euler Necessary Condition Section 21. Second Derivative of a Functional Section 22. Lagrange Conditions Section 23. Convex Functionals Section 24. Weak Convergence and Weak Compactness Section 25. Reflexive Spaces Section 26. Existence Theorems Section 27. Minimal Surfaces Section 28. Excursion on Numerical MethodsChapter V. The Topological Method Section 29. Existence Theorems Section 30. The Brouwer and the Leray-Schauder Degree of a Mapping Section 31. General Boundary Conditions for Second Order Ordinary Differential Equations Section 32. Summary of Chapters IV and V. Some Additional RemarksChapter VI. Noncoercive Problems Section 33. Vanishing Nonlinearities. Regular Case Section 34. Vanishing Nonlinearities. Singular Case Section 35. Jumping Nonlinearities with Finite Jumps Section 36. Jumping Nonlinearities with Infinite Jumps Section 37. Rapid Nonlinearities Section 38. Periodic ProblemsChapter VII. Variational Inequalities Section 39. Formulation of the Problem Section 40. More on the Definition of the Solution of a Variational Inequality Section 41. Examples Section 42. Some Special Results Section 43. Existence TheoremsReferencesIndex
