Math for CS Heat Equation. Laplace Equation. Tutorial 12 Heat Equation. Laplace Equation. 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 Tutorial 12

Contents Heat Equation Solution of the Heat Equation Math for CS Contents Heat Equation Solution of the Heat Equation Examples of the Physical Equations 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 Tutorial 12

Example: The Heat Equation Math for CS Example: The Heat Equation The heat equation, describing the temperature in solid u(x,y,z,t) as a function of position (x,y,z) and time t: This equation is derived as follows: Consider a small square of size δ, shown on the figure. Its heat capacitance is δ2·q, where q is the heat capacitance per unit area. The heat flow inside this square is the difference of the flows through its four walls. The heat flow through each wall is: y Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. x Math for CS Tutorial 12

Math for CS The Heat Equation Here δ is the size of the square, µ is the heat conductivity of the body and is the temperature gradient. The change of the temperature of the body is the total thermal flow divided by its heat capacitance: the last expression is actually the definition of the second derivative, therefore: 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 Tutorial 12

Solution of the Heat Equation 1/5 Math for CS Solution of the Heat Equation 1/5 A square plate [0,1]2 has the sides kept at u(0,y,t)=u(1,y,t)=u(x,0,t)=u(x,1,t)=0, and initial temperature u(x,y,0)=f(x,y). Determine u(x,y,t). Solution: Consider the solution in the form Then (1) becomes (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. (2) Math for CS Tutorial 12

Solution of the Heat Equation 2/5 Math for CS Solution of the Heat Equation 2/5 Since the left side depends only on t and the right side depends only on x and y, each side must be equal to a constant, -λ2, where the sign is chosen for convergence: The last equation can be rewritten as , which shows that both sides are constants, say -µ2 (negative to have bounded solutions): (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. (3) Math for CS Tutorial 12

Solution of the Heat Equation 3/5 Math for CS Solution of the Heat Equation 3/5 The solutions to equations (2) and (3) are: therefore, the solution to (1) is given by: From the boundary conditions that u(0,y,t)=0 and u(x,0,t)=0 we get a1=a2=0; Therefore the solution is already limited to the form: where B=b1b2. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (4) Math for CS Tutorial 12

Solution of the Heat Equation 4/5 Math for CS Solution of the Heat Equation 4/5 From the boundary conditions u(1,y,t)=0 and u(x,1,t)=0, we see that the sin(..x) and sin(..y) are zero at the boundary, therefore: Therefore, the solution satisfying boundary conditions is given by: Since the equation (1) is linear, any linear combination of these functions is also a solution. Letting t=0, and using an initial condition u(x,y,0)=f(x,y), 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. (5) Math for CS Tutorial 12

Solution of the Heat Equation 5/5 Math for CS Solution of the Heat Equation 5/5 The solution to (1) is given by Where And therefore 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 Tutorial 12

Math for CS Laplace Equation 1/3 Three sides of the square plate are kept at zero temperature u(0,y)=u(1,y)=u(x,0)=0, the fourth side, is kept at temperature u1: u(x,1)= u1. Determine the steady state temperature of the plate. To solve, suppose u(x,y)=X(x)Y(y): or Setting each side to –λ2 we obtain: From which 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 Tutorial 12

Laplace Equation 2/3 Where Math for CS Laplace Equation 2/3 Where The boundary conditions u(0,y)=u(x,0)=0 imply a1=a2=0. The condition u(1,y)=0 implies λ=mπ and therefore, the general form of the solution is: From the condition u(x,1)=u1 we have: And therefore 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 Tutorial 12

Laplace Equation 3/3 The solution is Math for CS Laplace Equation 3/3 The solution is This problem, which is the solution of Laplace equation Inside the region R when u(x,y) is specified an the boundary of R is called a Dirichlet problem. The boundary conditions, when the function is specified around the boundary is called Dirichlet boundary conditions. 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 Tutorial 12

Examples of Physical Equations Math for CS Examples of Physical Equations The vibrating string equation, describing the deviation y(x,t) of the taut string from its equilibrium y=0 position: The derivation of this equation is somewhat similar to the heat equation: we consider a small piece of the string; the force acting on this piece is ; it causes the acceleration of the piece which is . 3. The Schrödinger equation. This equation defines the wave function of the particle in the static field, and used, for example to calculate the electron orbits of the atoms. (10) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (11) Math for CS Tutorial 12

Example 1 (14) (15) (16) Solve Given Solution Math for CS Example 1 Solve Given Solution The solution consists of functions: The condition u(0,t)=u(3,t)=0 is fulfilled by (14) (15) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (16) Math for CS Tutorial 12

Math for CS Example 1 We need only n=12, 24 and 30 in order to fit f(x,0). The solution is (17) 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 Tutorial 12

Solution of the String Equation Math for CS Solution of the String Equation The vibrating string equation Can be solved in the way similar to solution of the heat equation. Substituting into (14), we obtain (18) 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 Tutorial 12

Math for CS Example 2 (1/3) The taut string equation is fixed at points x=-1 and x=1; f(-1,t)=f(1,t)=0; Its equation of motion is Initially it is pulled at the middle, so that Find out the motion of the string. Solution: The solution of (19) is comprised of the functions, obtaining zero values at x=-1 and x=1: (19) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (20) Math for CS Tutorial 12

Example 2 (2/3) (21) Solution (continued): Math for CS Example 2 (2/3) Solution (continued): Moreover, since the initial condition is symmetric, only the cos(…x) remains in the solution. The coefficients bn in (20) are zeros, since Therefore, the solution has the form (21) 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 Tutorial 12

Example 2 (3/3) Where Math for CS Tutorial 12 Math for CS 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 Tutorial 12