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Lecture 13: Sturm-Liouville Problem, Laplace Equation for 3D Sphere, Legendre Polynomials

This lecture covers the Sturm-Liouville problem, Laplace equation for a 3D sphere, and the use of Legendre polynomials in solving second-order PDEs.

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Lecture 13: Sturm-Liouville Problem, Laplace Equation for 3D Sphere, Legendre Polynomials

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  1. Lecture/Tutorial13 Contents Partial Differential Equations Sturm-Liuville Problem Laplace Equation for 3D sphere Legandre Polynomials Lecture 13

  2. 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) Lecture 13

  3. 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) Lecture 13

  4. 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. Lecture 13

  5. Examples Lecture 13

  6. 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: Lecture 13

  7. 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. Lecture 13

  8. 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 Lecture 13

  9. 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. Lecture 13

  10. 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 : (3) (4) Lecture 13

  11. 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: (5) Lecture 13

  12. 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) Lecture 13

  13. 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: (7) Lecture 13

  14. 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: Lecture 13

  15. 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. Lecture 13

  16. 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) Lecture 13

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