1. Lecture 13 Contents Partial Differential Equations
Math for CS
Lecture 13
Contents
Partial Differential Equations
Sturm-Liuville Problem
Laplace Equation for 3D sphere
Legandre Polynomials
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2. Math for CS
Second order PDE’s
The second order quasi-linear equation is defined by:
It is called ‘quasi-linear’ because the left hand side (LHS) is linear in the dependent variable, but the RHS function may not be. In the short-hand notation this equation looks:
(1)
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3. Hyperbolic, Parabolic and Elliptic PDE’s
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Hyperbolic, Parabolic and Elliptic PDE’s
This PDE can be hyperbolic, parabolic, or elliptic, depending on the sign of the term
B2-4AC (which can vary with x and y, if A, B, and C are not constants):
For the motivations of this notation, we consider the simple forms of each of 3 cases. For this let B=0 and RHS be constant C:
(2)
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Lecture 13

4. Motivation for the notation
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Motivation for the notation
The simplest solutions for these equations are
For more complex equation (1) the type change as a function of coordinate, however the local properties of the solution still depend on the sign of discrimintor B2-4AC.
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5. Examples Application Equation Coefficients B2-4AC Class Wave Equation
Math for CS
Examples
Application
Equation
Coefficients
B2-4AC
Class
Wave Equation
A=1,B=0,C=-α2
>0
Hyperbolic
Heat Equation
Parabolic
Poisson’s Equation
A=1,B=0,C=1
<0
Elliptic
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6. Special Coordinate systems
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Special Coordinate systems
When solving a second order PDE’s in special coordinate systems, the specific representation of Laplacian arises:
In cylindrical:
And in Spherical:
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7. Special Coordinate systems
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Special Coordinate systems
In these cases, the variable separation approach also facilitates the solution. In the Euclidian case the eigenfunctions were Fourier series. Here, after the substitution
The differential equations arise, which solutions are special functions like Legendre polinomials or Bessel functions.
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8. Sturm-Liuville Problem
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Sturm-Liuville Problem
The special functions, which arise in these homogeneous Boundary Value Problems (BVPs) with homogeneous boundary conditions (BCs) are mostly special cases of Sturm-Liouville Problem, given by:
On the interval a≤x≤b, with the homogeneous boundary conditions
The values λn, that yield the nontrivial solutions are called eigenvalues, and the corresponding solutions yn(x) are eigenfunctions.
The set of eigenfunctions, {yn(x)}, form an orthogonal system with respect to the weight function, p(x), over the interval.
If p(x), q(x), and r(x) are real, the eigenvalues are also real
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9. String Equation Consider the case r(x)=1, p(x)=1, q(x)=0:
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String Equation
Consider the case r(x)=1, p(x)=1, q(x)=0:
And boundary conditions y(0)=y(π)=0.
Case 1 - Negative Eigenvalues: For this case we try λ=-ν2. With this substitution, the original ODE becomes:
This is just a simple, constant coefficient, second-order ODE with characteristic equation
and roots
Thus, the general solution for the negative eigenvalue assumption is
The boundary conditions give:
Therefore, the nontrivial solution is only for non-negative eigenvalues, which is a familiar Fourier series.
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10. Steady State Temperature in a Sphere
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Steady State Temperature in a Sphere
Find the steady state temperature of a sphere of radius 1, when the temperature of upper half is held at T=100 and the lower half at T=0. Inside the sphere, the temperature satisfies the Laplace equation.
The Laplace equation in spherical coordinates is:
Substitute and multiply by :
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(3)
(4)
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Lecture 13

11. Steady State Temperature in a Sphere 2
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Steady State Temperature in a Sphere 2
If we multiply by sin2θ, the last term became a function of φ only, while the first two do not depend on φ, therefore, the last term is a constant. It must be negative, since the meaningful solutions must be 2π periodic.
Now the equation can be rewritten as:
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(5)
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12. Steady State Temperature in a Sphere 3
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Steady State Temperature in a Sphere 3
The first term is a function of r, while the last two are functions of θ, therefore:
Making the change x=cosθ, we obtain: dx=sin θdθ, and
(6)
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13. Steady State Temperature in a Sphere 4
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Steady State Temperature in a Sphere 4
This is called the equation for associated Legendre polynomials. It is in fact the specific case of Sturm-Liuville problem
When
It has a solutions for k=l(l+1), which is the Legandre’s polynoms:
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(7)
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Lecture 13

14. Steady State Temperature in a Sphere 5
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Steady State Temperature in a Sphere 5
The equation (6)
Has the solutions
However, the solution with negative degree is not physical, since it is singular in the center of the sphere. Combining all together into (4), we obtain:
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15. Steady State Temperature in a Sphere 6
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Steady State Temperature in a Sphere 6
Now, since the boundary condition does not depend on φ, the solution reduces to m=0:
The coefficients cl are determined to satisfy the boundary conditions at r=1:
,where f(x)=0, -1<x<0 and f(x)=1, 0<x<1.
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16. Steady State Temperature in a Sphere 7
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Steady State Temperature in a Sphere 7
,where f(x)=0, -1<x<0 and f(x)=1, 0<x<1. For calculation of cl, we use the Rodriges formula and normalization of Legendre’s polynomials (given here without proof)
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Lecture 13