First Order Partial Differential Equations, Part - 1:
Single Linear and Quasilinear First Order Equations
PHOOLAN PRASAD
DEPARTMENT OF MATHEMATICS
INDIAN INSTITUTE OF SCIENCE, BANGALORE
Definition
First order PDE in two independent variables is a relation
Introduction and Basic Definitions
First-Order Partial Differential Equations
Course plan
Lecture Slot:
Division I(CSE, ECE) and II (EEE, EPH, MA&C)-Slot C
Division III(ME, CE) and IV (BT, CL, CT)-Slot E
Tutorial Slot:-Slot E
Class Schedule
Slot C: Mon:10
MA 201: Lecture - 3
Solving quasilinear PDEs
Quasi-linear First-Order PDEs
A first order quasi-linear PDE is of the form
a(x, y , u)
u
u
+ b(x, y , u)
= c(x, y , u).
x
y
(1)
Quasi-linear First-Order PDEs
A first order quasi-linear PDE is of the form
a(x,
Surfaces orthogonal to a given system of surfaces
Nonlinear First-Order PDEs
Solving Cauchys problem for nonlinear PDEs
MA 201: Lecture - 4
Nonlinear first order PDES
IIT Guwahati
MA201(2017):PDE
Surfaces orthogonal to a given system of surfaces
Nonlinear
Classification of PDEs
Canonical forms
MA 201: Lecture - 7
Second order PDEs: Classification and Canonical
forms
MA201(2017):PDE
Classification of PDEs
Canonical forms
Classification of PDEs
An ODE is classified according to its order and whether it is
l
Compatible Systems and Charpits Method
Charpits Method
Some Special Types of First-Order PDEs
MA 201: Lecture - 5
Charpit method
IIT Guwahati
MA201(2017):PDE
Compatible Systems and Charpits Method
Charpits Method
Some Special Types of First-Order PDEs
Def
Jacobi Method
Second order PDE
MA 201: Partial Differential Equations
Jacobi method for I order PDE
and
Introduction to II order PDE
IIT Guwahati
MA201(2017):PDE
Jacobi Method
Second order PDE
Jacobis Method
Consider the following form of PDE
PDE :
f (x,
MA 201: Lecture - 2
Methods of characteristics
MA201(2017):PDE
Example
Let us warm up with a simple example
ux = u + c, c is function of x, y .
MA201(2017):PDE
(1)
Example
Let us warm up with a simple example
ux = u + c, c is function of x, y .
(1)
Observ
MA 201 MATHEMATICS III (PARTIAL DIFFERENTIAL EQUATIONS)
Topic-wise reference for first order PDE
S. No.
1
2
3
4
5
Topics
Introduction to PDE, formation of PDEs,
origin of PDEs
Integral surface passing through a given curve
(Cauchy Problem) for linear and
MA 201, Mathematics III, July-November 2016,
Fourier Transform (Contd.)
Lecture 17
Lecture 17
MA 201, PDE (2016)
1 / 26
Fourier Transform
For a given function f , the improper integral
Z
1
eit
2
Z
Z
1
1
f ( )ei d d =
eit g()d
2
2
is called Fourier int
MA 201: Partial Differential Equations
DAlemberts Solution Lecture - 7
MA 201 (2016), PDE
1 / 20
MA 201 (2016), PDE
2 / 20
Vibrating string and the wave equation
Consider a stretched string of length with the ends fastened to the ends x = 0
and x = .
Su
Motivating Examples
Compatible Systems and Charpits Method
Charpits Method
Some Special Types of First-Order PDEs
MA 201: Partial Differential Equations
Lecture - 5
IIT Guwahati
MA201(2016):PDE
Motivating Examples
Compatible Systems and Charpits Method
Ch
MA 201: Second Order Linear PDE
Canonical Transformation
Lecture - 6
MA201(2016):PDE
A second order PDE with two independent variables x and y is
given by
F (x, y , u, ux , uy , uxy , uxx , uyy ) = 0.
(1)
What is the linear form?
The unknown function u(
Quasi-linear First-Order PDEs
Formation of quasi-linear First-Order PDEs
General solution quasi-linear First-Order PDEs
The method of Lagrange
Solving a Cauchy Problem
MA 201: Partial Differential Equations
Lecture - 3
IIT Guwahati
MA201(2016):PDE
Quasi-l
Surfaces orthogonal to a given system of surfaces
Nonlinear First-Order PDEs
Solving Cauchys problem for nonlinear PDEs
MA 201: Partial Differential Equations
Lecture - 4
IIT Guwahati
MA201(2016):PDE
Surfaces orthogonal to a given system of surfaces
Nonli
MA 201: Convergence of Fourier Series
Lecture - 9
MA201(2016):PDE
Convergence of Fourier Series for Periodic Functions
Example
Recall that Fourier Series of the function f (x) = x for L x L is
given by
2L X
1
nx
(1)n+1 sin
.
n=1
n
L
Observe that value o
MA 201: Finite Vibrating string Problem
Introduction to Fourier Series
Lecture - 8
MA201(2016):PDE
IBVP for Vibrating string with no external forces
We consider the problem in a computational domain
(x, t) [0, L] [0, )
The IBVP under consideration consi
MA 101 (Mathematics I)
Sequence : Summary of Lectures
A sequence of real numbers or a sequence in R is a mapping f : N R.
Notation: We write xn for f (n), n N and so the notation for a sequence is (xn ).
Examples:
1. Constant sequence: (a, a, a, .), where
Riemann Integral: Motivation
Riemann Integral: Motivation
Partition of [a, b]: A nite set cfw_x0 , x1 , ., xn [a, b] such that
a = x0 < x1 < < xn = b.
Riemann Integral: Motivation
Partition of [a, b]: A nite set cfw_x0 , x1 , ., xn [a, b] such that
a =
An innite series in R is an expression
n=1
where (xn ) is a sequence in R.
xn ,
An innite series in R is an expression
xn ,
n=1
where (xn ) is a sequence in R.
More formally, it is an ordered pair (xn ), (sn ),
where (xn ) is a sequence in R,
and sn = x1
A sequence of real numbers or a sequence in R
is a mapping f : N R.
A sequence of real numbers or a sequence in R
is a mapping f : N R.
Notation: We write xn for f (n), n N
and so the notation for a sequence is (xn ).
A sequence of real numbers or a seque
Differentiability and Derivative: Let D R and let x0 D such
that there exists an interval I of R satisfying x0 I D.
Differentiability and Derivative: Let D R and let x0 D such
that there exists an interval I of R satisfying x0 I D.
A function f : D R is s
MA 101 (Mathematics - I)
Innite Series : Summary of Lectures
An innite series in R is an expression
xn , where (xn ) is a sequence in R.
n=1
More formally, it is an ordered pair (xn ), (sn ), where (xn ) is a sequence in R and
sn = x1 + + xn for all n N.
MA 101 (Mathematics I)
Integration : Summary of Lectures
Riemann Integral: Motivation
Partition of [a, b]: A nite set cfw_x0 , x1 , ., xn [a, b] such that a = x0 < x1 < < xn = b.
Upper sum & Lower sum: Let f : [a, b] R be bounded. For a partition P = cfw
Denition: Let D(= ) R and let f : D R.
Denition: Let D(= ) R and let f : D R.
We say that f is continuous at x0 D if for each > 0, there
exists > 0 such that |f (x) f (x0 )| < for all x D satisfying
|x x0 | < .
Denition: Let D(= ) R and let f : D R.
We sa
MA 101 (Mathematics I)
Dierentiation : Summary of Lectures
Dierentiability and Derivative: Let D R and let x0 D such that there exists
an interval I of R satisfying x0 I D.
(x
A function f : D R is said to be dierentiable at x0 if lim f (x)f 0 0 ) (or equ
Chapter 24
Logistic Regression
Content list
Purpose of logistic regression
Assumptions of logistic regression
The logistic regression equation
Interpreting log odds and odds ratio
Model t and likelihood function
SPSS activity a logistic regression analysi
Logistic Regression
Chapter 8
Aims
When and Why do we Use Logistic
Regression?
Binary
Multinomial
Theory Behind Logistic Regression
Assessing the Model
Assessing predictors
Things that can go Wrong
Interpreting Logistic Regression
Slide 2
When And