Math for CSLecture 131 Contents Partial Differential Equations Sturm-Liuville Problem Laplace Equation for 3D sphere Legandre Polynomials Lecture/Tutorial13

Math for CSLecture 132 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)

Math for CSLecture 133 Hyperbolic, Parabolic and Elliptic PDE’s This PDE can be hyperbolic, parabolic, or elliptic, depending on the sign of the term B 2 -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)

Math for CSLecture 134 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 B 2 -4AC.

Math for CSLecture 135 Examples ApplicationEquationCoefficientsB 2 -4ACClass Wave EquationA=1,B=0,C=-α 2 >0Hyperbolic Heat EquationA=1,B=0,C=-α 2 0Parabolic Poisson’s EquationA=1,B=0,C=1<0Elliptic

Math for CSLecture 136 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:

Math for CSLecture 137 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.

Math for CSLecture 138 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 λ n, that yield the nontrivial solutions are called eigenvalues, and the corresponding solutions y n (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

Math for CSLecture 139 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=-ν 2. 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.

Math for CSLecture 1310 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)

Math for CSLecture 1311 Steady State Temperature in a Sphere 2 If we multiply by sin 2 θ, 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)

Math for CSLecture 1312 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)

Math for CSLecture 1313 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)

Math for CSLecture 1314 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:

Math for CSLecture 1315 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 c l are determined to satisfy the boundary conditions at r=1:,where f(x)=0, -1<x<0 and f(x)=1, 0<x<1.

Math for CSLecture 1316 Steady State Temperature in a Sphere 7,where f(x)=0, -1<x<0 and f(x)=1, 0<x<1. For calculation of c l, we use the Rodriges formula and normalization of Legendre’s polynomials (given here without proof)