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Nonlinear boundary value problems for ordinary differential equations

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Granas A., Guenther R., Lee J.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Comments............................................................................................................................5 CHAPTER I Introduction § 1. Elementary theory of second order differential equations...........................................12 § 2. Topological preliminaries.............................................................................................14 § 3. The maximum principle................................................................................................16 § 4. Existence and a priori bounds-examples.....................................................................19 § 5. Problems with other boundary conditions....................................................................25 CHAPTER II The Bernstein theory of the equation y" = f(t, y, y') § 1. The homogeneous Dirichlet, Neumann, and periodic problems...................................28 § 2. The homogeneous Sturm-Liouville problem................................................................34 § 3. Inhomogeneous boundary conditions..........................................................................35 § 4. Examples and remarks................................................................................................39 § 5. Bernstein-Nagumo growth conditions..........................................................................44 § 6. Nonlinear boundary conditions....................................................................................50 § 7. Uniqueness..................................................................................................................52 CHAPTER III Applications § 1. Steady-state temperature distributions........................................................................56 § 2. The Thomas-Fermi problem........................................................................................59 § 3. Singular boundary value problems..............................................................................62 § 4. Osmotic flow.................................................................................................................64 § 5. Positive solutions to diffusion equations......................................................................70 CHAPTER IV Other second order boundary value problems § 1. Periodic solutions to differential equations of Nirenberg type......................................76 § 2. The Dirichlet problem for y" = f(y') and the Neumann problem for y" = f(t,y,y').............85 § 3. Upper and lower solutions...........................................................................................94 CHAPTER V Even order systems and higher order equations § 1. General existence theorems........................................................................................99 § 2. Second order systems...............................................................................................102 § 3. Third and fourth order problems................................................................................108 § 4. Higher even order equations......................................................................................111 CHAPTER VI Numerical solution of boundary value problems § 1. Newton’s method........................................................................................................113 § 2. The shooting method for the Dirichlet problem..........................................................115 § 3. The shooting method for the Neumann problem........................................................120 § 4. Quasilinearization for boundary value problems........................................................121 References.......................................................................................................................125

Ograniczanie wyników

1 Granas A.

1 Guenther R.

1 Lee J.

1 1985

Wyniki wyszukiwania

Nonlinear boundary value problems for ordinary differential equations