Analytical solution of PDEs
From charlesreid1
Part of the CFD lecture set.
See also Courant Hilbert I: Section 5
Contents
Introduction
Most methods for analytically solving PDEs transform them into systems of ODEs (ordinary differential equations). A fairly comprehensive list of techniques might include:
- Separation of variables - reduces a PDE of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} independent variables into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} ODEs
- Integral transforms - reduce a PDE of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} variables into a PDE of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n-1} variables (so, useful for 2-variable PDEs)
- Integral equations - changes a PDE into an integral equation, solved using other techniques
- Change of coordinates - changes a PDE into an ODE (or, into an easier PDE) by changing the problem coordinates
- Dependent variable transforms - transform the PDE unknown into a new, easier-to-find unknown
- Perturbation methods - changes a nonlinear problem into a sequence of linear problems that can approximate the nonlinear problem
- Impulse-response technique - decomposes initial and boundary conditions of the problem into simple impulses to find the response to each impulse; the final response is the sum of the simple impulse responses
- Eigenfunction expansion - finds solutions of the PDE in the form of infinite sums of eigenfunctions
- Calculus of variations - reformulates equations into minimization problems, e.g. total energy minimization
Separation of Variables
Restrictions:
- Linear PDEs
- Homogeneous PDEs
- Boundary conditions of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha u_{x}(0,t) + \beta u (0,t) = 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma u_{x}(1,t) + \delta u(1,t) = 0} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha, \beta, \gamma, \delta} are constants (making the boundary conditions linear and homogeneous).
The principle behind separation of variables is to find solutions of the form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{n}(x,t) = X_{n}(x) T_{n}(t)}
or,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{n}(x,t) = X_{n}(x) + T_{n}(t)}
In principle, an infinite number of solutions exist which will satisfy the PDE and also satisfy the boundary conditions (hence the subscript Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} ).
These solutions are used as building blocks, and the sum of all of these solutions gives the total solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(x,t)} .
The idea behind separation of variables (and the reason it only applies to homogeneous problems) is, if one has an equation of the form
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial X}{\partial x} \frac{\partial T}{\partial t} = 0 }
or,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial X}{\partial x} + \frac{\partial T}{\partial t} = 0 }
then each of the terms must be equal to a constant - otherwise the derivative of X would create a function in x that could not be canceled out by the other derivatives.
Given these infinite solutions, and given the boundary conditions, the solution can be found by adding up all the simple solutions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_n (x) T_n(t)} in a way that also matches the initial conditions:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty}} A_n X_n (x) T_n(t) }
or,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \displaystyle{ \sum_{n=1}^{\infty} } A_n (X_n (x) + T_n (t) ) }
Example 1: First Derivatives
This example uses separation of variables to solve the PDE:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{x} \times u_{y} = 0}
This is a straightforward application of separation of variables. A solution of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(x,y) = X(x) \times Y(y)} will be assumed. Plugging this in:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial X}{\partial x} \times \frac{\partial Y}{\partial y} = 0 }
Which means that each derivative must be constant - otherwise (as mentioned above) there would be functions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x,y,z} left over:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d X}{d x} = c_1 }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d Y}{d y} = c_2 }
So, the 1st order PDE has been transformed into a set of 2 ODEs. These are easy to solve, and yield Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X = c_1 x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Y = c_2 y} , so the solution is:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(x,y) = ( c_1 x ) ( c_2 y ) = c_3 x y }
Which, when plugged in, satisfies the original PDE.
Example 2: Heat Equation
Separation of variables can be used to solve the heat equation under certain circumstances
What circumstances?
linear homogeneous boundary conditions... linear homogeneous PDE
Consider the heat equation in a normalized domain:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta_{t} = \alpha^2 \theta_{xx} \qquad 0 < x < 1, 0 < t < \infty }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta} stands for temperature.
Given the (linear homogeneous) boundary conditions
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle BCs = \begin{cases} \theta(0,t) = 0 & 0 < t < \infty \\ \theta(1,t) = 0 & 0 < t < \infty \end{cases} }
Now we will use this PDE to describe the temperature profile in a finite rod, with the (relative) temperature at either end fixed at zero, and some initial temperature profile - which is arbitrary - let's call it Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(x)} .
Is it homogeneous?
Is this PDE linear? What if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha = \alpha( \theta )} ? Can we still use separation of variables?
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha(\theta)} makes the equation quasi-linear - i.e. NOT LINEAR - so can't use the separation of variables (see Introduction to partial differential equations lecture)
Step 1: Solution to PDE
A solution of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(x) T(t)} can be assumed, and plugged into the PDE:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(x) T'(t) = \alpha^2 X''(x) T(t) }
and, dividing each side by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(x) T(t)} , it becomes
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{T'(t)}{\alpha^2 T(t)} = \frac{X''(x)}{X(x)} }
which must be equal to a constant, since both expressions are functions of different variables.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{T'(t)}{\alpha^2 T(t)} = \frac{X''(x)}{X(x)} = c }
Both sides can be written as ODEs - so the PDE was transformed into two ODEs:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} T' - k \alpha^2 T &=& 0 \\ X'' - kX &=& 0 \end{align} }
Once we solve these ODEs, the solutions can be multiplied together to give the solution to the PDE.
NOTE: Assuming the initial temperature profile is higher than 0 requires Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k} to be negative - otherwise the temporal behavior of the solution at infinite times would be to increase (not physically possible). It can be replaced with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k = -\lambda^2} , which is guaranteed to be negative.
Thus the ODE system is:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} T' + \lambda^2 \alpha^2 T &=& 0 \\ X'' + \lambda^2 X &=& 0 \end{align} }
General solutions can be found for both PDEs, of the form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} T(t) &=& c_1 \exp{( - \lambda^2 \alpha^2 t )} \\ X(x) &=& c_2 sin(\lambda x) + c_3 cos( \lambda x ) \end{align} }
So the infinite solutions to our PDE are:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x,t) = \exp{( -\lambda^2 \alpha^2 t )} \left( A sin(\lambda x) + B cos(\lambda x) \right) }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A,B,\lambda} are arbitrary.
Step 2: Finding solution to PDE and BCs
Let's try these solutions at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t=0} - our initial condition
Try the function for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A=1,B=1,\lambda=1} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x=0} :
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \theta(0,0) &=& \exp{( -\lambda^2 \alpha^2 t )} \left[ A sin(\lambda x) + B cos(\lambda x) \right] \\ &=& \exp{( - 1^2 \times \alpha^2 \times 0 )} \left[ (1) sin( 1 \times 0 ) + (1) cos( 1 \times 0) \right] \\ &=& 1 \times ( 0 + 1 ) \\ &=& 1 \end{array} }
But our boundary condition says Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x=0,t) = 0} !
Is the problem with the function?
No, problem is with the values of the constants we chose
We need to find values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A, B, \lambda} that will match boundary conditions
Boundary condition 1:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x=0,t) = 0 }
This means:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \theta(x=0,t) &=& \exp{( -\lambda^2 \alpha^2 t )} \left[ A sin( \lambda \times 0 ) + B cos( \lambda \times 0 ) \right] \\ \theta(x=0,t) &=& B \exp{( -\lambda^2 \alpha^2 t )} \\ 0 &=& B \exp{( -\lambda^2 \alpha^2 t )} \end{array} }
Which means either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} is 0, or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \exp{( -\lambda^2 \alpha^2 t )}} is 0.
When is the exponential function 0?
Never - so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B=0} always, in order to make this true.
Now let's do the same thing for the other boundary condition:
Boundary condition 2:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x=1,t) = 0 }
What do you think we're going to find? Will we learn something about the exponential term? No - it's a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t} , and it's never going to be zero anyway.
Will we find something out from the cosine term? No - we know it's gone away permanently.
So we're going to get some information from the sine term.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \theta(x=1,t) &=& \exp{( -\lambda^2 \alpha^2 t )} \left[ A sin( \lambda \times 1 ) \right] \\ 0 &=& A sin( \lambda ) \exp{( -\lambda^2 \alpha^2 t )} \end{array} }
Okay! So, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A=0} ! All done! That was easy!
Trivial solution - Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A=0} is an example of a trivial solution. It means our initial condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(x) = 0} , so we started out with a uniform temperature of 0.
This leaves one non-zero term:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle sin(\lambda) = 0 }
which tells us something about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} - not about A!
We have to pick functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta = X(x) T(t)} such that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda = n \pi }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} is an integer.
This means we've found an infinite number of functions that satisfy the PDE and the boundary conditions:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x,t) = A_n \exp{\left( -(n\pi\alpha)^2 t \right)} sin(n \pi x) }
So the solution to the PDE will be a sum of these simple functions.
Step 3: Solution to PDE, BCs, ICs
The last remaining step is to find a sum of the fundamental solutions, which satisfy the PDE and BCs, that will also satisfy the initial condition. That is, find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_n} in
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x,t) = \sum_{n=1}^{\infty} A_n \exp{\left( -(n\pi\alpha)^2 t \right)} sin(n \pi x) }
such that it satisfies the initial condition,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x,t=0) = \phi(x) }
This raises a question asked by Fourier: can an arbitrary function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(x)} be represented as an infinite series of sine functions?
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(x,t=0) = \phi(x) = \sum_{n=1}^{\infty} A_n sin( n \pi x ) }
(see wikipedia:Fourier series and wikipedia:Fourier transform).
To find values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_n} , we can use a property called orthogonality.
Orthogonality means that the integral of two functions that are orthogonal is 0.
For sine functions, any sine functions that are not the same are orthogonal, i.e.:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{1} sin( m \pi x ) sin( n \pi x ) dx = \begin{cases} 0 & m \neq n \\ \frac{1}{2} & m = n \end{cases} }
(Convince yourself of this by studying the plots of several sine waves:)
Starting with the expression for the initial condition:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(x) = A_1 sin(\pi x) + A_2 sin(2 \pi x) + \dots }
Next, multiplying both sides by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle sin(m \pi x)} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m} an arbitrary integer, and integrating from 0 to 1:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \int_{0}^{1} \phi(x) sin(m \pi x) dx &=& A_m \int_{0}^{1} sin^2 (m \pi x) dx \\ &=& \frac{1}{2} A_m \end{array} }
where all other terms are zero due to orthogonality.
This gives an expression for the coefficients of the Fourier series,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_mm = 2 \int_{0}^{1} \phi(x) sin( m \pi x ) dx }
Making the final solution:
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta(u,x) = \sum_{n=1}^{\infty} A_n \exp{( -(n \pi \alpha)^2 t )} sin(n \pi x) } |
with the coefficients given by
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_n = 2 \int_{0}^{1} \phi(x) sin(n \pi x) dx } |
How many people were expecting a different solution? How many are disappointed by this result?
Since we put all this effort into finding this solution, I want to spend some time figuring out what it means.
Let's look at a hypothetical initial temperature profile:
This temperature profile plotted above (for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha = 0.01} ) is an expression consisting of multiple sine waves, each with a unique frequency. The full expression for the temperature profile is:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{array}{l} \theta(x,t) = 25 \exp{( - \pi^2 \alpha^2 t )} sin( \pi x ) \\ -4 \exp{( - (2)^2 \pi^2 \alpha^2 t )} sin( 2 \pi x ) \\ +15 \exp{( - (3)^2 \pi^2 \alpha^2 t )} sin( 3 \pi x ) \\ +4 \exp{( - (4)^2 \pi^2 \alpha^2 t )} sin( 4 \pi x ) \\ +8 \exp{( - (5)^2 \pi^2 \alpha^2 t )} sin( 5 \pi x ) \\ + \exp{( - (8)^2 \pi^2 \alpha^2 t )} sin( 8 \pi x ) \\ +2 \exp{( -(20)^2 \pi^2 \alpha^2 t )} sin(20 \pi x ) \\ +0.5\exp{( -(21)^2 \pi^2 \alpha^2 t )} sin(21 \pi x ) \\ + \exp{( -(25)^2 \pi^2 \alpha^2 t )} sin(25 \pi x ) \\ + \exp{( -(28)^2 \pi^2 \alpha^2 t )} sin(28 \pi x ) \\ + \exp{( -(35)^2 \pi^2 \alpha^2 t )} sin(35 \pi x ) \\ +0.5\exp{( -(100)^2 \pi^2 \alpha^2 t )} sin(100 \pi x ) \end{array} }
Now, notice that each term has the decaying exponential function (the time function).
For higher frequencies, this exponential function will decay more rapidly. For lower frequencies, it will decay slowly.
This means the high-frequency fluctuations will decay very quickly...
And the low-frequency fluctuations will decay more slowly.
Plotting this solution function for various times yields a time-dependent solution for this function:
Notice how the small-scale (high-frequency) fluctuations are dampened immediately, whereas the moderate length-scale (middle-frequency) fluctuations take longer to be dampened, and finally, the last frequency to be dampened is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=1} frequency (half a sine wave).
This mimics the natural behavior of diffusion of heat - small-scale temperature fluctuations will be dampened first, followed by increasingly larger scale temperature fluctuations.
Combination of Variables
Courant Hilbert II:
Section 3 Part 1 (p.40 of PDF)
Method of Characteristics
References
http://www.exampleproblems.com/wiki/index.php/PDE:Integration_and_Separation_of_Variables
Tannehill
Anderson
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