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 Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

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) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Hyperbolic, Parabolic and Elliptic PDE’s Math for CS 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) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Motivation for the notation Math for CS 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. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

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 Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Special Coordinate systems Math for CS 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: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Special Coordinate systems Math for CS 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. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Sturm-Liuville Problem Math for CS 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 Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

String Equation Consider the case r(x)=1, p(x)=1, q(x)=0: Math for CS 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. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Steady State Temperature in a Sphere Math for CS 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 : Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (3) (4) Math for CS Lecture 13

Steady State Temperature in a Sphere 2 Math for CS 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: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (5) Math for CS Lecture 13

Steady State Temperature in a Sphere 3 Math for CS 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) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Steady State Temperature in a Sphere 4 Math for CS 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: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (7) Math for CS Lecture 13

Steady State Temperature in a Sphere 5 Math for CS 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: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Steady State Temperature in a Sphere 6 Math for CS 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. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13

Steady State Temperature in a Sphere 7 Math for CS 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) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 13