MA 201: Partial Differential Equations Lecture - 7 · MA 201: Partial Differential Equations...
Transcript of MA 201: Partial Differential Equations Lecture - 7 · MA 201: Partial Differential Equations...
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
MA 201: Partial Differential EquationsLecture - 7
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• A 2nd order PDE with two independent variables x and y is given by
F (x , y , u, ux , uy , uxy , uxx , uyy ) = 0. (1)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• A 2nd 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?
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• A 2nd 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 = u(x , y) satisfies an equation:
Auxx + Buxy + Cuyy + Dux + Euy + Fu + G = 0. (2)
Coefficients and G are sufficiently smooth functions of x and y .
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• A 2nd 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 = u(x , y) satisfies an equation:
Auxx + Buxy + Cuyy + Dux + Euy + Fu + G = 0. (2)
Coefficients and G are sufficiently smooth functions of x and y .
Facts:
• Introducing the differential operators X = ∂∂x
and Y = ∂∂y
, equation
(2) can be written as
(AX 2 + BXY + CY 2 + DX + EY + FI )u = G . (3)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• A 2nd 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 = u(x , y) satisfies an equation:
Auxx + Buxy + Cuyy + Dux + Euy + Fu + G = 0. (2)
Coefficients and G are sufficiently smooth functions of x and y .
Facts:
• Introducing the differential operators X = ∂∂x
and Y = ∂∂y
, equation
(2) can be written as
(AX 2 + BXY + CY 2 + DX + EY + FI )u = G . (3)
• Classification of such PDEs is based on expression
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• A 2nd 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 = u(x , y) satisfies an equation:
Auxx + Buxy + Cuyy + Dux + Euy + Fu + G = 0. (2)
Coefficients and G are sufficiently smooth functions of x and y .
Facts:
• Introducing the differential operators X = ∂∂x
and Y = ∂∂y
, equation
(2) can be written as
(AX 2 + BXY + CY 2 + DX + EY + FI )u = G . (3)
• Classification of such PDEs is based on expression
Lu = Auxx + Buxy + Cuyy ,
which is called the Principal part of the equation.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• At a point (x , y), the above equation is said to be
Hyperbolic if B2(x , y)− 4A(x , y)C (x , y) > 0
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• At a point (x , y), the above equation is said to be
Hyperbolic if B2(x , y)− 4A(x , y)C (x , y) > 0
Parabolic if B2(x , y)− 4A(x , y)C (x , y) = 0
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• At a point (x , y), the above equation is said to be
Hyperbolic if B2(x , y)− 4A(x , y)C (x , y) > 0
Parabolic if B2(x , y)− 4A(x , y)C (x , y) = 0
Elliptic if B2(x , y) − 4A(x , y)C (x , y) < 0
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• At a point (x , y), the above equation is said to be
Hyperbolic if B2(x , y)− 4A(x , y)C (x , y) > 0
Parabolic if B2(x , y)− 4A(x , y)C (x , y) = 0
Elliptic if B2(x , y) − 4A(x , y)C (x , y) < 0
• Each category relates to specific problems
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• At a point (x , y), the above equation is said to be
Hyperbolic if B2(x , y)− 4A(x , y)C (x , y) > 0
Parabolic if B2(x , y)− 4A(x , y)C (x , y) = 0
Elliptic if B2(x , y) − 4A(x , y)C (x , y) < 0
• Each category relates to specific problems
1. An Elliptic Equation
uxx + uyy = 0.
Which is known ‘Laplace’s Equation’.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• At a point (x , y), the above equation is said to be
Hyperbolic if B2(x , y)− 4A(x , y)C (x , y) > 0
Parabolic if B2(x , y)− 4A(x , y)C (x , y) = 0
Elliptic if B2(x , y) − 4A(x , y)C (x , y) < 0
• Each category relates to specific problems
1. An Elliptic Equation
uxx + uyy = 0.
Which is known ‘Laplace’s Equation’.
2. A Hyperbolic Equation
utt − uxx = 0.
Which is known as ‘Wave’ or ‘Vibrating String Equation’.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
• At a point (x , y), the above equation is said to be
Hyperbolic if B2(x , y)− 4A(x , y)C (x , y) > 0
Parabolic if B2(x , y)− 4A(x , y)C (x , y) = 0
Elliptic if B2(x , y) − 4A(x , y)C (x , y) < 0
• Each category relates to specific problems
1. An Elliptic Equation
uxx + uyy = 0.
Which is known ‘Laplace’s Equation’.
2. A Hyperbolic Equation
utt − uxx = 0.
Which is known as ‘Wave’ or ‘Vibrating String Equation’.
3. A parabolic Equation
ut = uxx .
Which is known as ‘Heat Equation’.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibration and Wave
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibration and Wave
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibration and Wave
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String ProblemConsider a stretched string of
length π with the ends fastenedto the ends x = 0 and x = π.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String ProblemConsider a stretched string of
length π with the ends fastenedto the ends x = 0 and x = π.
Suppose that the string is setto vibrate by displacing it fromits equilibrium position.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String ProblemConsider a stretched string of
length π with the ends fastenedto the ends x = 0 and x = π.
Suppose that the string is setto vibrate by displacing it fromits equilibrium position.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String ProblemConsider a stretched string of
length π with the ends fastenedto the ends x = 0 and x = π.
Suppose that the string is setto vibrate by displacing it fromits equilibrium position.
• Let u(x , t) denote thetransverse displacement attime t ≥ 0 of the point onthe string at position x .
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String ProblemConsider a stretched string of
length π with the ends fastenedto the ends x = 0 and x = π.
Suppose that the string is setto vibrate by displacing it fromits equilibrium position.
• Let u(x , t) denote thetransverse displacement attime t ≥ 0 of the point onthe string at position x . Thatis, we assume that each pointof the string moves only inthe vertical direction.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String ProblemConsider a stretched string of
length π with the ends fastenedto the ends x = 0 and x = π.
Suppose that the string is setto vibrate by displacing it fromits equilibrium position.
• Let u(x , t) denote thetransverse displacement attime t ≥ 0 of the point onthe string at position x . Thatis, we assume that each pointof the string moves only inthe vertical direction.
• In particular, u(x , 0) denotesthe initial shape of the stringand ut(x , 0) denotes theinitial velocity.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String ProblemConsider a stretched string of
length π with the ends fastenedto the ends x = 0 and x = π.
Suppose that the string is setto vibrate by displacing it fromits equilibrium position.
• Let u(x , t) denote thetransverse displacement attime t ≥ 0 of the point onthe string at position x . Thatis, we assume that each pointof the string moves only inthe vertical direction.
• In particular, u(x , 0) denotesthe initial shape of the stringand ut(x , 0) denotes theinitial velocity.
• That is, the string is set tovibrate by supplying an initialvelocity ut(x , 0) to it from itsequilibrium position u(x , 0)and then releasing it.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String Problem Contd..
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String Problem Contd..
• Under some physical assumptions, we arrive at the following equation
utt(x , t) = c2uxx (x , t), (4)
which governs the entire process.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String Problem Contd..
• Under some physical assumptions, we arrive at the following equation
utt(x , t) = c2uxx (x , t), (4)
which governs the entire process.
• Here, c represents a physical quantity.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String Problem Contd..
• Under some physical assumptions, we arrive at the following equation
utt(x , t) = c2uxx (x , t), (4)
which governs the entire process.
• Here, c represents a physical quantity.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String Problem Contd..
• Under some physical assumptions, we arrive at the following equation
utt(x , t) = c2uxx (x , t), (4)
which governs the entire process.
• Here, c represents a physical quantity.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Vibrating String Problem Contd..
• Under some physical assumptions, we arrive at the following equation
utt(x , t) = c2uxx (x , t), (4)
which governs the entire process.
• Here, c represents a physical quantity.
• Equation (4) is also well-known as Wave Equation.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider following wave equation
utt(x , t) = c2uxx (x , t). (5)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider following wave equation
utt(x , t) = c2uxx (x , t). (5)
• It is easy to recognize that above equation reduces to followingequations
( ∂
∂t+ c
∂
∂x
)( ∂
∂t− c
∂
∂x
)
u = 0
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider following wave equation
utt(x , t) = c2uxx (x , t). (5)
• It is easy to recognize that above equation reduces to followingequations
( ∂
∂t+ c
∂
∂x
)( ∂
∂t− c
∂
∂x
)
u = 0 or (6)
( ∂
∂t− c
∂
∂x
)( ∂
∂t+ c
∂
∂x
)
u = 0. (7)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider following wave equation
utt(x , t) = c2uxx (x , t). (5)
• It is easy to recognize that above equation reduces to followingequations
( ∂
∂t+ c
∂
∂x
)( ∂
∂t− c
∂
∂x
)
u = 0 or (6)
( ∂
∂t− c
∂
∂x
)( ∂
∂t+ c
∂
∂x
)
u = 0. (7)
• So, we try to solve individual first order equations, that is
ut + cux = 0
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider following wave equation
utt(x , t) = c2uxx (x , t). (5)
• It is easy to recognize that above equation reduces to followingequations
( ∂
∂t+ c
∂
∂x
)( ∂
∂t− c
∂
∂x
)
u = 0 or (6)
( ∂
∂t− c
∂
∂x
)( ∂
∂t+ c
∂
∂x
)
u = 0. (7)
• So, we try to solve individual first order equations, that is
ut + cux = 0 and (8)
ut − cux = 0. (9)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
whose characteristic equations are
dt
1=
dx
c=
du
0.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
whose characteristic equations are
dt
1=
dx
c=
du
0.
Considering 1st and 2nd pair, we obtain
x − ct = c1.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
whose characteristic equations are
dt
1=
dx
c=
du
0.
Considering 1st and 2nd pair, we obtain
x − ct = c1.
Then, due to Lagrange theorem any arbitrary function ofφ = x − ct will be the solution to the PDE (10).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
whose characteristic equations are
dt
1=
dx
c=
du
0.
Considering 1st and 2nd pair, we obtain
x − ct = c1.
Then, due to Lagrange theorem any arbitrary function ofφ = x − ct will be the solution to the PDE (10). We assume that
u = f (φ) = f (x − ct)
is a solution for the PDE (10),
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
whose characteristic equations are
dt
1=
dx
c=
du
0.
Considering 1st and 2nd pair, we obtain
x − ct = c1.
Then, due to Lagrange theorem any arbitrary function ofφ = x − ct will be the solution to the PDE (10). We assume that
u = f (φ) = f (x − ct)
is a solution for the PDE (10), where f is an arbitrary function of φ.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
whose characteristic equations are
dt
1=
dx
c=
du
0.
Considering 1st and 2nd pair, we obtain
x − ct = c1.
Then, due to Lagrange theorem any arbitrary function ofφ = x − ct will be the solution to the PDE (10). We assume that
u = f (φ) = f (x − ct)
is a solution for the PDE (10), where f is an arbitrary function of φ.
• Similarly, it is an easy exercise to verify that u = g(x + ct) is thesolution for the PDE ut − cux = 0,
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation• Consider equation
ut + cux = 0, (10)
whose characteristic equations are
dt
1=
dx
c=
du
0.
Considering 1st and 2nd pair, we obtain
x − ct = c1.
Then, due to Lagrange theorem any arbitrary function ofφ = x − ct will be the solution to the PDE (10). We assume that
u = f (φ) = f (x − ct)
is a solution for the PDE (10), where f is an arbitrary function of φ.
• Similarly, it is an easy exercise to verify that u = g(x + ct) is thesolution for the PDE ut − cux = 0, where g is an arbitrary functionof x + ct.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..• We are in a process of solving wave equation
utt(x , t)− c2uxx = 0. (11)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..• We are in a process of solving wave equation
utt(x , t)− c2uxx = 0. (11)
• In this process, we have found that u = f (x − ct) is a solution tothe PDE
ut + cux = 0 (12)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..• We are in a process of solving wave equation
utt(x , t)− c2uxx = 0. (11)
• In this process, we have found that u = f (x − ct) is a solution tothe PDE
ut + cux = 0 (12)
• And u = g(x + ct) is a solution for the PDE
ut − cux = 0, (13)
where f is an arbitrary function of x − ct and g is an arbitraryfunction of x + ct.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..• We are in a process of solving wave equation
utt(x , t)− c2uxx = 0. (11)
• In this process, we have found that u = f (x − ct) is a solution tothe PDE
ut + cux = 0 (12)
• And u = g(x + ct) is a solution for the PDE
ut − cux = 0, (13)
where f is an arbitrary function of x − ct and g is an arbitraryfunction of x + ct.
• Then, observe that u(x , t) = f (x − ct) + g(x + ct) is a solution tothe equation (11).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..• We are in a process of solving wave equation
utt(x , t)− c2uxx = 0. (11)
• In this process, we have found that u = f (x − ct) is a solution tothe PDE
ut + cux = 0 (12)
• And u = g(x + ct) is a solution for the PDE
ut − cux = 0, (13)
where f is an arbitrary function of x − ct and g is an arbitraryfunction of x + ct.
• Then, observe that u(x , t) = f (x − ct) + g(x + ct) is a solution tothe equation (11). In fact, it is the general solution for the waveequation.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..• We are in a process of solving wave equation
utt(x , t)− c2uxx = 0. (11)
• In this process, we have found that u = f (x − ct) is a solution tothe PDE
ut + cux = 0 (12)
• And u = g(x + ct) is a solution for the PDE
ut − cux = 0, (13)
where f is an arbitrary function of x − ct and g is an arbitraryfunction of x + ct.
• Then, observe that u(x , t) = f (x − ct) + g(x + ct) is a solution tothe equation (11). In fact, it is the general solution for the waveequation.
• The physical interpretations of the functions f and g are quiteinteresting.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
Then observe that
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
Then observe that
• At time t = t1, position of a object traveling with the wave is givenby
(x , y) = (x , sin(x − ct1)).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
Then observe that
• At time t = t1, position of a object traveling with the wave is givenby
(x , y) = (x , sin(x − ct1)).
• Location of the object at time t = 0 and x = 0 is (0, 0).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
Then observe that
• At time t = t1, position of a object traveling with the wave is givenby
(x , y) = (x , sin(x − ct1)).
• Location of the object at time t = 0 and x = 0 is (0, 0).
• Location of the object at time t = t1 and x = ct1 is (ct1, 0).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
Then observe that
• At time t = t1, position of a object traveling with the wave is givenby
(x , y) = (x , sin(x − ct1)).
• Location of the object at time t = 0 and x = 0 is (0, 0).
• Location of the object at time t = t1 and x = ct1 is (ct1, 0).
• In time t1 the object has moved from 0 to ct1 horizontally.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
For instance, consider
y(x , t) = sin(x − ct).
Then observe that
• At time t = t1, position of a object traveling with the wave is givenby
(x , y) = (x , sin(x − ct1)).
• Location of the object at time t = 0 and x = 0 is (0, 0).
• Location of the object at time t = t1 and x = ct1 is (ct1, 0).
• In time t1 the object has moved from 0 to ct1 horizontally. So, cdenotes the velocity of the wave propagation.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
Thus, we observe that
• The function f (x − ct) defines a wave propagation moving at aconstant speed c in the positive direction of the x-axis.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
Thus, we observe that
• The function f (x − ct) defines a wave propagation moving at aconstant speed c in the positive direction of the x-axis.
• Similarly the function g(x + ct) defines a wave progressing in thenegative direction of the x-axis with constant velocity c .
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
Thus, we observe that
• The function f (x − ct) defines a wave propagation moving at aconstant speed c in the positive direction of the x-axis.
• Similarly the function g(x + ct) defines a wave progressing in thenegative direction of the x-axis with constant velocity c .
• The solutionu(x , t) = f (x − ct) + g(x + ct)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
Thus, we observe that
• The function f (x − ct) defines a wave propagation moving at aconstant speed c in the positive direction of the x-axis.
• Similarly the function g(x + ct) defines a wave progressing in thenegative direction of the x-axis with constant velocity c .
• The solutionu(x , t) = f (x − ct) + g(x + ct)
to the wave equation
utt − c2uxx = 0
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Solution for Wave Equation Contd..
Thus, we observe that
• The function f (x − ct) defines a wave propagation moving at aconstant speed c in the positive direction of the x-axis.
• Similarly the function g(x + ct) defines a wave progressing in thenegative direction of the x-axis with constant velocity c .
• The solutionu(x , t) = f (x − ct) + g(x + ct)
to the wave equation
utt − c2uxx = 0
is, therefore, the algebraic sum of these two traveling waves.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert Solution For Wave Equation
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert Solution For Wave Equation
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert Solution For Wave Equation
We consider the following initialvalue problem
utt − c2uxx = 0, x ∈ (−∞,∞), t > 0 (14)
Displacement: u(x , 0) = φ(x) (15)
Velocity: ut(x , 0) = ψ(x) .(16)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert Solution For Wave Equation
We consider the following initialvalue problem
utt − c2uxx = 0, x ∈ (−∞,∞), t > 0 (14)
Displacement: u(x , 0) = φ(x) (15)
Velocity: ut(x , 0) = ψ(x) .(16)
From the solution u(x , t) = f (x − ct) + g(x + ct) to the wave equation(14), we find that
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert Solution For Wave Equation
We consider the following initialvalue problem
utt − c2uxx = 0, x ∈ (−∞,∞), t > 0 (14)
Displacement: u(x , 0) = φ(x) (15)
Velocity: ut(x , 0) = ψ(x) .(16)
From the solution u(x , t) = f (x − ct) + g(x + ct) to the wave equation(14), we find that
f (x) + g(x) = φ(x),
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert Solution For Wave Equation
We consider the following initialvalue problem
utt − c2uxx = 0, x ∈ (−∞,∞), t > 0 (14)
Displacement: u(x , 0) = φ(x) (15)
Velocity: ut(x , 0) = ψ(x) .(16)
From the solution u(x , t) = f (x − ct) + g(x + ct) to the wave equation(14), we find that
f (x) + g(x) = φ(x), (17)
−cf ′(x) + c g ′(x) = ψ(x), (18)
for all values of x .
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert solution of the wave equation (Contd...)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert solution of the wave equation (Contd...)Integrating the second equation with respect to x
−f (x) + g(x) =1
c
∫ x
0
ψ(τ)dτ + A. (19)
where A = g(0)− f (0).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert solution of the wave equation (Contd...)Integrating the second equation with respect to x
−f (x) + g(x) =1
c
∫ x
0
ψ(τ)dτ + A. (19)
where A = g(0)− f (0). Again, we have f (x) + g(x) = φ(x).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert solution of the wave equation (Contd...)Integrating the second equation with respect to x
−f (x) + g(x) =1
c
∫ x
0
ψ(τ)dτ + A. (19)
where A = g(0)− f (0). Again, we have f (x) + g(x) = φ(x). Whichleads to
f (x) =1
2
[
φ(x)−1
c
∫ x
0
ψ(τ)dτ
]
− A/2, (20)
g(x) =1
2
[
φ(x) +1
c
∫ x
0
ψ(τ)dτ
]
+ A/2, (21)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert solution of the wave equation (Contd...)Integrating the second equation with respect to x
−f (x) + g(x) =1
c
∫ x
0
ψ(τ)dτ + A. (19)
where A = g(0)− f (0). Again, we have f (x) + g(x) = φ(x). Whichleads to
f (x) =1
2
[
φ(x)−1
c
∫ x
0
ψ(τ)dτ
]
− A/2, (20)
g(x) =1
2
[
φ(x) +1
c
∫ x
0
ψ(τ)dτ
]
+ A/2, (21)
Substituting these expressions into u(x , t) = f (x − ct) + g(x + ct), wehave
u(x , t) =1
2
[
φ(x − ct) + φ(x + ct)]
+1
2c
∫ x+ct
x−ct
ψ(τ)dτ. (22)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
The D’Alembert solution of the wave equation (Contd...)Integrating the second equation with respect to x
−f (x) + g(x) =1
c
∫ x
0
ψ(τ)dτ + A. (19)
where A = g(0)− f (0). Again, we have f (x) + g(x) = φ(x). Whichleads to
f (x) =1
2
[
φ(x)−1
c
∫ x
0
ψ(τ)dτ
]
− A/2, (20)
g(x) =1
2
[
φ(x) +1
c
∫ x
0
ψ(τ)dτ
]
+ A/2, (21)
Substituting these expressions into u(x , t) = f (x − ct) + g(x + ct), wehave
u(x , t) =1
2
[
φ(x − ct) + φ(x + ct)]
+1
2c
∫ x+ct
x−ct
ψ(τ)dτ. (22)
This is D’Alembert’s (1717-1783) solution of one-dimensional wave
equation, which he actually derived around 1746.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (23)
with IC’s u(x , 0) = 0, ut(x , 0) = 0 ∀x ∈ (−∞,∞). (24)
Due to the presence of F in the right side of the equation (35),
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (23)
with IC’s u(x , 0) = 0, ut(x , 0) = 0 ∀x ∈ (−∞,∞). (24)
Due to the presence of F in the right side of the equation (35), wedirectly can not apply D’Alembert’s (1717-1783) solution.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (23)
with IC’s u(x , 0) = 0, ut(x , 0) = 0 ∀x ∈ (−∞,∞). (24)
Due to the presence of F in the right side of the equation (35), wedirectly can not apply D’Alembert’s (1717-1783) solution. Rather, werelate the given IVP (23)-(24) with a homogeneous IVP
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (23)
with IC’s u(x , 0) = 0, ut(x , 0) = 0 ∀x ∈ (−∞,∞). (24)
Due to the presence of F in the right side of the equation (35), wedirectly can not apply D’Alembert’s (1717-1783) solution. Rather, werelate the given IVP (23)-(24) with a homogeneous IVP by convertingF (x , t) into a initial function.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (23)
with IC’s u(x , 0) = 0, ut(x , 0) = 0 ∀x ∈ (−∞,∞). (24)
Due to the presence of F in the right side of the equation (35), wedirectly can not apply D’Alembert’s (1717-1783) solution. Rather, werelate the given IVP (23)-(24) with a homogeneous IVP by convertingF (x , t) into a initial function.
More precisely, we consider an user defined problem
vtt − c2vxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (25)
with IC’s v(x , 0) = 0, vt(x , 0) = F (x , ξ), ξ > 0. (26)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Non-Homogeneous Wave Equation: Duhamel’s Principle
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (23)
with IC’s u(x , 0) = 0, ut(x , 0) = 0 ∀x ∈ (−∞,∞). (24)
Due to the presence of F in the right side of the equation (35), wedirectly can not apply D’Alembert’s (1717-1783) solution. Rather, werelate the given IVP (23)-(24) with a homogeneous IVP by convertingF (x , t) into a initial function.
More precisely, we consider an user defined problem
vtt − c2vxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (25)
with IC’s v(x , 0) = 0, vt(x , 0) = F (x , ξ), ξ > 0. (26)
Then, Duhamel’s Principle yields the solution u from v
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Duhamel’s Principle Contd..
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Duhamel’s Principle Contd..Now, we have a user defined problem
vtt − c2vxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (27)
with IC’s v(x , 0) = 0, vt(x , 0) = F (x , ξ), ξ > 0. (28)
Note that the solution v of the above problem depends on x , t and ξ.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Duhamel’s Principle Contd..Now, we have a user defined problem
vtt − c2vxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (27)
with IC’s v(x , 0) = 0, vt(x , 0) = F (x , ξ), ξ > 0. (28)
Note that the solution v of the above problem depends on x , t and ξ.Thus, v = v(x , t, ξ).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Duhamel’s Principle Contd..Now, we have a user defined problem
vtt − c2vxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (27)
with IC’s v(x , 0) = 0, vt(x , 0) = F (x , ξ), ξ > 0. (28)
Note that the solution v of the above problem depends on x , t and ξ.Thus, v = v(x , t, ξ). Accordingly, we can modify the ICs as
v(x , 0, ξ) = 0, vt(x , 0, ξ) = F (x , ξ), ξ > 0.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Duhamel’s Principle Contd..Now, we have a user defined problem
vtt − c2vxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (27)
with IC’s v(x , 0) = 0, vt(x , 0) = F (x , ξ), ξ > 0. (28)
Note that the solution v of the above problem depends on x , t and ξ.Thus, v = v(x , t, ξ). Accordingly, we can modify the ICs as
v(x , 0, ξ) = 0, vt(x , 0, ξ) = F (x , ξ), ξ > 0.
Due to D’Alembert’s, v = v(x , t, ξ) is given by
v(x , t, ξ) =1
2c
∫ x+ct
x−ct
F (τ, ξ)dτ. (29)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Duhamel’s Principle Contd..Now, we have a user defined problem
vtt − c2vxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (27)
with IC’s v(x , 0) = 0, vt(x , 0) = F (x , ξ), ξ > 0. (28)
Note that the solution v of the above problem depends on x , t and ξ.Thus, v = v(x , t, ξ). Accordingly, we can modify the ICs as
v(x , 0, ξ) = 0, vt(x , 0, ξ) = F (x , ξ), ξ > 0.
Due to D’Alembert’s, v = v(x , t, ξ) is given by
v(x , t, ξ) =1
2c
∫ x+ct
x−ct
F (τ, ξ)dτ. (29)
Then the solution u is given by
u(x , t) =
∫ t
0
v(x , t − τ, τ)dτ
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Leibniz Rule for Differentiation
d
dt
(
∫ b(t)
a(t)
G (x , t)dx)
=
∫ b(t)
a(t)
Gtdx + G (b(t), t)b′(t)
−G (a(t), t)a′(t). (30)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Leibniz Rule for Differentiation
d
dt
(
∫ b(t)
a(t)
G (x , t)dx)
=
∫ b(t)
a(t)
Gtdx + G (b(t), t)b′(t)
−G (a(t), t)a′(t). (30)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Leibniz Rule for Differentiation
d
dt
(
∫ b(t)
a(t)
G (x , t)dx)
=
∫ b(t)
a(t)
Gtdx + G (b(t), t)b′(t)
−G (a(t), t)a′(t). (30)
Use Leibniz rule on
u(x , t) =
∫ t
0
v (x , t − τ, τ)dτ, v (x , 0, ξ) = 0, vt(x , 0, ξ) = F (x , ξ), ξ > 0,
to have
ut =
∫ t
0
vt(x , t − τ, τ)dτ + v(x , t − t, t)×dt
dt− v(x , t − 0, 0)× 0
=
∫ t
0
vt(x , t − τ, τ)dτ + v(x , 0, t) =
∫ t
0
vt(x , t − τ, τ)dτ.
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Leibniz Rule for Differentiation
d
dt
(
∫ b(t)
a(t)
G (x , t)dx)
=
∫ b(t)
a(t)
Gtdx + G (b(t), t)b′(t)
−G (a(t), t)a′(t). (30)
Use Leibniz rule on
u(x , t) =
∫ t
0
v (x , t − τ, τ)dτ, v (x , 0, ξ) = 0, vt(x , 0, ξ) = F (x , ξ), ξ > 0,
to have
ut =
∫ t
0
vt(x , t − τ, τ)dτ + v(x , t − t, t)×dt
dt− v(x , t − 0, 0)× 0
=
∫ t
0
vt(x , t − τ, τ)dτ + v(x , 0, t) =
∫ t
0
vt(x , t − τ, τ)dτ.
Simillarly
utt(x , t) =
∫ t
0
vtt (x , t − τ, τ)dτ + vt(x , 0, t) =
∫ t
0
vtt(x , t − τ, τ)dτ + F (x , t)
c2uxx(x , t) = c2
∫ t
0
vxx (x , t − τ, τ)dτ so that utt − c2uxx = F (x , t).
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Completely Non-Homogeneous Wave Equation: Duhamel’sPrinciple
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Completely Non-Homogeneous Wave Equation: Duhamel’sPrinciple
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (31)
with IC’s u(x , 0) = φ(x), ut(x , 0) = ψ(x) ∀x ∈ (−∞,∞). (32)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Completely Non-Homogeneous Wave Equation: Duhamel’sPrinciple
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (31)
with IC’s u(x , 0) = φ(x), ut(x , 0) = ψ(x) ∀x ∈ (−∞,∞). (32)
Try to break the problem as (ρ, θ) problems as: Find θ such that
θtt(x , t)− c2θxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (33)
with IC’s θ(x , 0) = 0, θt(x , 0) = 0 ∀x ∈ (−∞,∞). (34)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
-
Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Completely Non-Homogeneous Wave Equation: Duhamel’sPrinciple
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (31)
with IC’s u(x , 0) = φ(x), ut(x , 0) = ψ(x) ∀x ∈ (−∞,∞). (32)
Try to break the problem as (ρ, θ) problems as: Find θ such that
θtt(x , t)− c2θxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (33)
with IC’s θ(x , 0) = 0, θt(x , 0) = 0 ∀x ∈ (−∞,∞). (34)
Find ρ such that
ρtt(x , t)− c2ρxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (35)
with IC’s ρ(x , 0) = φ(x), ρt(x , 0) = ψ(x) ∀x ∈ (−∞,∞). (36)
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
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Second Order Linear PDEWave Equation
Method of Characteristics for Wave EquationInfinite String Problem
Completely Non-Homogeneous Wave Equation: Duhamel’sPrinciple
Find u(x , t) such that
utt(x , t)− c2uxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (31)
with IC’s u(x , 0) = φ(x), ut(x , 0) = ψ(x) ∀x ∈ (−∞,∞). (32)
Try to break the problem as (ρ, θ) problems as: Find θ such that
θtt(x , t)− c2θxx = F (x , t), (x , t) ∈ (−∞,∞)× (0,∞) (33)
with IC’s θ(x , 0) = 0, θt(x , 0) = 0 ∀x ∈ (−∞,∞). (34)
Find ρ such that
ρtt(x , t)− c2ρxx = 0, (x , t) ∈ (−∞,∞)× (0,∞) (35)
with IC’s ρ(x , 0) = φ(x), ρt(x , 0) = ψ(x) ∀x ∈ (−∞,∞). (36)
Principle of Super Position tells that ρ+ θ solve the completelynon-homogeneous problem
IIT Guwahati Dr Bhupen Deka, Department of Mathematics, IIT Guwahati, 2020
Second Order Linear PDEWave EquationMethod of Characteristics for Wave EquationInfinite String Problem