Introduction 1 Ordinary Differential Equations(ODE) 1.1 Linear System of Differential Equations 1.2 Fundamental Matrix of Solution 1.2.1 Properties 1.2.2 Constant Variation Formula 1.3 Exercises and Some Problem Solutions 1.3.1 Linear Constant Coefficients Equations 1.3.2 Some Stability Problems and Their Solution 1.4 Nonlinear Systems of Differential Equations 1.4.1 Existence and Uniqueness ofC.P(f,x0,y0) 1.4.2 Differentiability of Solutions with Respect to Parameters 1.4.3 The Local Flow(Differentiability Properties) 1.4.4 Applications(Using Differentiability of a Flow) 1.5 Gradient Systems of Vector Fields and Solutions 1.5.1 Gradient System Associated to Finite Set of Vec-Fields 1.5.2 Frobenius Theorem 1.6 Appendix 2 First Order PDE 2.1 Cauchy Problem for the Hamilton-Jacobi Equations 2.2 Nonlinear First Order PDE 2.2.1 Examples of Scalar NonlinearODE 2.2.2 Nonlinear Hamilton-Jacobi E quations 2.2.3 Exercises 2.3 Stationary Solutions for Nonlinear Ist Order PDE 2.3.1 Introduction 2.3.2 The Lie Algebra of Characteristic Fields 2.3.3 Parameterized Stationary Solutions 2.3.4 The linear Case:H0(x,p)= 2.3.5 The Case H0(x,p,u)=H0(x,p); Stationary Solutions 2.3.6 Some Problems 2.4 Overdetermined System of 1st O.PDE & Lie Algebras 2.4.1 Solution for Linear Homogeneous Over Determined System. 2.5 Nonsingular Representation of a Gradient System 2.6 1st Order Evolution System of PDE and C-K Theorem 2.6.1 Cauchy-Kowalevska Theorem 2.6.2 1st Order Evolution System of Hyperbolic & Elliptic Equations 2.7 (E3) Plate Equations 2.7.1 Exercises 2.7.2 The Abstract Cauchy-Kawalewska Theorem 2.8 Appendix Infinitesimal Invariance 2.8.1 Groups and Differential Equations 2.8.2 Prolongation 2.8.3 Systems of Differential Equations 2.8.4 Prolongation of Group Action and Vector Fields 2.8.5 Higher Order Equation 3 Second Order PDE 3.1 Introduction 3.2 Poisson Equation 3.3 Exercises 3.4 Maximum Principle for Harmonic Functions 3.4.1 The Wave Equation;Kirchhoff,D'Alembert & Poisson Formulas 3.5 Exercises 3.5.1 System of Hyperbolic and Elliptic Equations(Definition) 3.6 Adjoint 2nd order operator;Green formulas 3.6.1 Green Formula and Applications 3.6.2 Applications of Riemann Function 3.6.3 Green function for Dirichlet problem 3.7 Linear Parabolic Equations 3.7.1 The Unique Solution of the C.P (3.120) and (3.121) 3.7.2 Exercises 3.7.3 Maximum Principle for Heat Equation 3.8 Weak Solutions(Generalized Solutions) 3.8.1 Boundary Parabolic Problem 3.8.2 Boundary Hyperbolic Problem 3.8.3 Fourier Method, Exercises 3.9 Some Nonlinear Elliptic and Parabolic PDE 3.9.1 Nonlinear Parabolic Equation 3.9.2 Some Nonlinear Elliptic Equations 3.10 Exercises 3.11 Appendix I 3.12 Appendix II Variational Method Involving PDE 3.12.1 Introduction 3.12.2 E-L Equation for Distributed Parameters Functionals 3.12.3 Examples of PDE Involving E-L Equation 3.13 Appendix III Harmonic Functions 3.13.1 Harmonic Functions 3.13.2 Green Formulas 3.13.3 Recovering a Harmonic Function Inside a Ball I 3.13.4 Recovering a Harmonic Function Inside a Ball II 4 Stochastic Differential Equations 4.1 Properties of continuous semimartingales 4.2 Approximations of the diffusion equations 4.3 Stochastic rule of derivation 4.4 Appendix 4.4.1 Introduction 4.4.2 Some problems and their solutions 4.4.3 Solution for the Problem (P1) 4.4.4 Solution for the Problem (P2) 4.4.5 Multiple vector fields case 4.4.6 Solution for (P1) 4.4.7 Solution for (P2)
