Chapter 1: Introduction Department of Electrical Engineering - PowerPoint PPT Presentation

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary Chapter 1: Introduction Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw September 11, 2013 DE Lecture 1 王奕翔 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary 1 Definitions and Terminology Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 2 Initial-Value/Boundary-Value Problems 3 Mathematical Modeling with Differential Equations 4 Summary DE Lecture 1 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 1 Definitions and Terminology Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 2 Initial-Value/Boundary-Value Problems 3 Mathematical Modeling with Differential Equations 4 Summary DE Lecture 1 王奕翔

Definitions and Terminology An equation containing the derivatives of one or more dependent Identify the independent and dependent variables in the above two DE’s. Exercise dy Example Initial-Value/Boundary-Value Problems variables, with respect to one or more independent variables, is called a differential equation (DE). Definition (Differential Equations) Differential Equations Solutions to Differential Equations Classification of Differential Equations Definition of Differential Equations Summary Mathematical Modeling with Differential Equations DE Lecture 1 dx = xy , where y = f ( x ) is a function of x . ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 , where u = φ ( x , y ) is a function of x and y . 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 1 Definitions and Terminology Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 2 Initial-Value/Boundary-Value Problems 3 Mathematical Modeling with Differential Equations 4 Summary DE Lecture 1 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations Classification of DE By type By order By linearity DE Lecture 1 王奕翔

Definitions and Terminology Examples: Example: Involving partial derivatives Partial Differential Equation dz dx Initial-Value/Boundary-Value Problems dy DE Lecture 1 Involving ordinary derivatives Ordinary Differential Equation Type of Differential Equations Solutions to Differential Equations Classification of Differential Equations Definition of Differential Equations Summary Mathematical Modeling with Differential Equations Only 1 independent variable 2+ independent variables ∂ 2 u ∂ x 2 + ∂ 2 u dx = 3 xy 2 / z dx = xy , dy ∂ y 2 = 0 王奕翔

Definitions and Terminology Example (ODE) = Example (PDE) = dx Initial-Value/Boundary-Value Problems DE Lecture 1 Order: the highest derivative of the equation. Order of Differential Equations Mathematical Modeling with Differential Equations Summary Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations ) 3 d 2 y ( dy dx 2 + 5 − 4 y = e x ⇒ order is 2 . x ∂ 2 u ∂ x ∂ y = uy 2 ⇒ order is 2 . 王奕翔

Definitions and Terminology following general form : The coefficients a i ’s are only function of x , not y . (1) Hence a linear ODE can be written more explicitly as follows: Definition (Linear ODE) Initial-Value/Boundary-Value Problems F DE Lecture 1 Definition of Differential Equations Linearity of Differential Equations Mathematical Modeling with Differential Equations Summary Solutions to Differential Equations Classification of Differential Equations Every ODE of a function y = f ( x ) with order n can be written in the ( x , y , y ′ , . . . , y ( n ) ) = 0 . ⇒ F is linear in { y , y ′ , . . . , y ( n ) } . The ODE is linear ⇐ dx n + a n − 1 ( x ) d n − 1 y a n ( x ) d n y dx n − 1 + · · · + a 1 ( x ) dy dx + a 0 ( x ) y = g ( x ) . 王奕翔

Definitions and Terminology A: Not necessarily. dx dy dx dy Initial-Value/Boundary-Value Problems dx dy Example (Nonlinear ODE) DE Lecture 1 Classification of Differential Equations Mathematical Modeling with Differential Equations Summary Definition of Differential Equations Discussion Linearity of Differential Equations Solutions to Differential Equations If both y = f 1 ( x ) and y = f 2 ( x ) satisfy (1), does a linear combination of f 1 and f 2 satisfy (1)? d 2 y ( ) dx 2 + 5 − 4 y = e x is linear d 2 y ( ) dx 2 + 5 − 4 y = e y is nonlinear ) 2 d 2 y ( dx 2 + 5 − 4 y = e x is nonlinear 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 1 Definitions and Terminology Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 2 Initial-Value/Boundary-Value Problems 3 Mathematical Modeling with Differential Equations 4 Summary DE Lecture 1 王奕翔

Definitions and Terminology Example y . dy solutions Initial-Value/Boundary-Value Problems y . DE Lecture 1 Definition Summary Explicit vs. Implicit Solutions Solutions to Differential Equations Classification of Differential Equations Definition of Differential Equations Mathematical Modeling with Differential Equations Explicit solution: solutions can be expressed explicitly as y = φ ( x ) . Implicit solution: solutions in the form of a relation G ( x , y ) = 0 . dx = − x Consider the following ODE dy √ √ 1 − x 2 and y = φ 2 ( x ) = − 1 − x 2 are explicit Both y = φ 1 ( x ) = The relation x 2 + y 2 − 1 = 0 is an implicit solution. Because x 2 + y 2 − 1 = 0 = ⇒ 2 xdx + 2 ydy = 0 = ⇒ dx = − x 王奕翔

Definitions and Terminology Trivial Solutions Initial-Value/Boundary-Value Problems dy Consider the ODE Example A: Solutions to Differential Equations Mathematical Modeling with Differential Equations Classification of Differential Equations DE Lecture 1 Definition of Differential Equations Summary dx = x √ y . Verify that both y = x 4 16 and y = 0 are solutions. y = x 4 dx = x 3 x 4 /16 = x · x 2 /4 = x 3 4 ; x √ y = x √ ⇒ dy 16 = 4 Hence y = x 4 16 is a solution. Also, trivially y = 0 is a solution. We call y = 0 a trivial solution . 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary 1 Definitions and Terminology Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 2 Initial-Value/Boundary-Value Problems 3 Mathematical Modeling with Differential Equations 4 Summary DE Lecture 1 王奕翔

Definitions and Terminology Because all parallel lines have the same slope. solution. Initial-Value/Boundary-Value Problems In real applications, we need some conditions to specify a unique A differential equation usually has more than one solution. Initial-Value Problems Summary Mathematical Modeling with Differential Equations DE Lecture 1 dx = 1 . We can derive a family of solutions: For example, consider dy { y = x + c , c ∈ R } . The initial value is one of them(not necessarily at x = 0 ): y (0) = 2 = ⇒ c = 0 + 2 = 2 = ⇒ unique solution: y = x + 2 y (2) = − 1 = ⇒ c = − 1 − 2 = − 3 = ⇒ unique solution: y = x − 3 . 王奕翔

Definitions and Terminology Usually a n -th order ODE requires n initial/boundary conditions to Initial-Value/Boundary-Value Problems specify an unique solution. Example Fact (Number of Initial/Boundary Conditions) Number of Initial/Boundary Conditions vs. Order Summary Mathematical Modeling with Differential Equations DE Lecture 1 For the ODE y ′′ = 2 , the family of solutions take the form y = x 2 + bx + c . Initial condition: y (1) = − 1 , y ′ (1) = 3 = ⇒ b = 1 , c = − 3 . Boundary condition: y (0) = 3 , y ′ (1) = 3 = ⇒ b = 1 , c = − 3 . 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary Some Remarks on Initial/Boundary Conditions Remark (Initial vs. Boundary Conditions) Boundary Conditions: conditions can be at different x . Remark (Order of the derivatives in the conditions DE Lecture 1 Initial Conditions: all conditions are at the same x = x 0 . Initial/boundary conditions can be the value or the function of 0 -th to ( n − 1) -th order derivatives, where n is the order of the ODE. 王奕翔

Definitions and Terminology Initial-Value/Boundary-Value Problems Mathematical Modeling with Differential Equations Summary 1 Definitions and Terminology Definition of Differential Equations Classification of Differential Equations Solutions to Differential Equations 2 Initial-Value/Boundary-Value Problems 3 Mathematical Modeling with Differential Equations 4 Summary DE Lecture 1 王奕翔