Partial Differential Equations III/IV
[6pt] Exercise Sheet 6
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1.
The Fourier transform: The heat equation with source term.
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(i)
Verify that
satisfies the ODE
This is an example of Duhamel’s principle, which is a method for obtaining a solution of an inhomogeneous differential equation, in this case , from the corresponding homogeneous differential equation, in this case .
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(ii)
Consider the heat equation on with source term :
Use the Fourier transform and part (i) to derive the solution
where is the fundamental solution of the heat equation in .
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(i)
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2.
The Fourier transform: The transport equation.
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(i)
Let and define by , which is the translation of by . Use a change of variables to prove that
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(ii)
Use the Fourier transform and part (i) to derive the solution of the transport equation
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(i)
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3.
The Fourier transform: Schrödinger’s equation. Consider Schrödinger’s equation
(1) where and are complex-valued.
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(i)
Use the Fourier transform to derive the solution
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(ii)
Let satisfy (1). We write to denote the function for fixed . Assume that and that for all . Use the Fourier transform to prove that
This can also be proved using the energy method.
Hint: Use the fact that the Fourier transform preserves the –norm: and (you do not need to prove this).
Remark: This shows that the energy is constant. Recall that if satisfies the heat equation on with diffusion constant , then the energy decays:Schrödinger’s equation is an example of a dispersive equation, where energy is conserved, whereas the heat equation is an example of a diffusion equation, where energy decays.
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(i)
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4.
The Fourier transform: The wave equation. Use the Fourier transform to derive the solution
of the wave equation
where the constant is the wave speed. This is known as D’Alembert’s solution.
Hint: Use Q2(i) and the fact that . -
5.
The Fourier transform of a derivative. Let . Use integration by parts to prove that
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6.
The Fourier transform of a convolution. Let . Prove that
Hint: By definition
The trick is to write
and then to interchange the order of integration.
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7.
Proof of the Sobolev embedding using the Fourier transform. In this question we use the Fourier transform to give an alternative proof of the Sobolev embedding for the case . Assume that , , and . Recall that
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(i)
Prove that
Hint: Use the fact that the Fourier transform preserves the –norm: for all (you do not need to prove this).
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(ii)
Prove that there exists a constant such that
Hint: Write
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(iii)
Use the Fourier Inversion Theorem to prove that
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(i)
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8.
Fundamental Solution of the Heat Equation. The Fundamental Solution of the Heat Equation in is
Verify that satisfies the heat equation
for all , .
Remark: It can be shown that as in the sense of distributions. -
9.
Finite speed of propagation for a degenerate diffusion equation. Define by
Let , where is a constant.
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(i)
Show that satisfies the degenerate diffusion equation
for all , , except for
where it is not differentiable. (It can also be shown that satisfies the the degenerate diffusion equation in all of in a suitable weak sense.)
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(ii)
Show that the map has compact support for all . Therefore, unlike the heat/diffusion equation, the degenerate diffusion equation has finite speed of propagation.
Remark: Observe that the diffusion coefficient vanishes when . Compare this to the case of the heat equation, where is strictly positive. For the 4H students: Just like for the Fundamental Solution of the Heat Equation, it can be shown that as in the sense of distributions.
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(i)
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10.
The mathematical equation that caused the banks to crash. The Black-Scholes PDE, or “the mathematical equation that caused the banks to crash” (Ian Stewart, The Observer, 12 Feb 2012), is the parabolic PDE
where is the price of a European option as a function of the stock price at time , is the risk-free interest rate, and is the volatility of the stock (see Wikipedia https://en.wikipedia.org/wiki/Black-Scholes_equation). Consider the change of variables
where , , are constants. Show that the Black-Scholes PDE is the heat equation in disguise:
So the heat equation is “the mathematical equation that caused the banks to crash”!
Remark: You can read Ian Stewart’s article here: https://www.theguardian.com/science/2012/feb/12/black-scholes-equation-credit-crunch -
11.
The energy method: Uniqueness for the heat equation in a time dependent domain. Let , be constants. Let be smooth functions with for all . Let be the non-cylindrical domain
Consider the heat equation
Use the energy method to prove that this equation has at most one smooth solution.
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12.
The energy method: Uniqueness for a 4th-order heat equation. Let , . Prove that there exists at most one smooth solution of the 4th-order heat equation
Since the equation is 4th-order, we prescribe boundary conditions on both and . Why do we consider to be the 4th-order version of the heat equation instead of , which at first sight seems to be closer to the heat equation ?
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13.
Asymptotic behaviour of the heat equation with time independent data. Let be open, bounded and connected with smooth boundary. Let be a smooth function satisfying
where , , are given smooth functions. Let be a smooth, time independent solution of the same equation:
Define . Use the energy method to prove that in as . In other words, if the source term and boundary data are independent of time, then the solution of the heat equation converges to the solution of Poisson’s equation in the –norm as .
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14.
Asymptotic behaviour of the heat equation with time independent data in the –norm. Let be a constant and let be a smooth function satisfying the heat equation
where and are smooth functions. Let be the unique solution of
Define .
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(i)
Prove that satisfies
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(ii)
Prove that in as .
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(iii)
Prove that in as .
Hint: Show that satisfies a heat equation. -
(iv)
Prove that in as .
Hint: By part (i), -
(v)
Conclude that in as .
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(i)
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15.
Applications of the maximum principle: Uniqueness and bounds on solutions. This question appeared on the May 2012 exam, Q9(b),(c).
Given , let with and let .-
(i)
Show that in there exists at most one solution to the problem
with on the parabolic boundary .
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(ii)
Assume that is a solution to the problem in (i). Show that we have
for .
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(i)
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16.
Application of the maximum principle: Comparison Principle. For , let be a smooth function satisfying
where are given smooth functions. Assume that , , and . Prove that in .
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17.
Eigenfunctions of the Laplacian and an application to the heat equation. Let , , be eigenvalue-eigenfunction pairs for on with zero Dirichlet boundary conditions, which means that and that satisfies
In Exercise Sheet 4 we used the energy method to show that and for all . By relabelling if necessary, we can assume that (in fact it can be shown that . Let satisfy the heat equation
Roughly speaking, it can be shown that the set of eigenfunctions forms a basis for the vector space of smooth functions on that vanish on . By writing and with respect to this basis as
show formally that
(2) Remark: From expression (2) we see that the rate of convergence of to as depends on the smallest eigenvalue of . This should not come as a surprise: When we proved that as using the energy method, we saw that the rate of convergence depends on the Poincaré constant (see Q13), and from Exercise Sheet 4, Q12 we know that the optimal Poincaré constant depends on the smallest eigenvalue of .