1. Differential Equations (ODE and PDE)

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1 1. Differential Equations (ODE and PDE) 1.1. Ordinary Differential Equations (ODE) So far we have dealt with Ordinary Differential Equations (ODE): involve derivatives with respect to only one variable if the sought after function depends on more than one variable, the extra variables were treated as constants! examples u (t) = λu(t) (radioactive decay) ( ) p (t) = r 1 p(t) p(t) (population model, logistic growth) k (ṗ(t) ) ( ) f(p(t), q(t)) p(t) = (predator-prey) q(t) g(p(t), q(t)) q(t) usually, initial conditions were required Page 1 of 19

2 1.2. (PDE) involve (partial) derivatives with respect to more than one variable, and the sought after function depends on more than one variable boundary conditions and/or initial conditions required (depending on type of PDE) Examples: 1D heat equation: 2 u(x, t) = u(x, t) t x2 also written as u t (x, t) = u xx (x, t) 2D Laplace equation: 2 2 u(x, y) + u(x, y) = x2 y2 also written as u xx (x, y) + u yy (x, y) = Page 2 of 19

3 2. choosing PDE instead of ODE in modelling is often a result of adding spatial dimensions: in population modelling: ODE-modelling assumes homogeneous distribution of the population in space switch to PDE if assumption is no longer applicable heat conduction: localising hot spots requires spatial resolution (i.e. PDE) time is often treated as a special dimension, we distinguish stationary problems: no time-dependence unsteady/instationary problems: time-dependence (perhaps, but not necessarily, with a stationary limit) Page 3 of 19

4 Heat Conduction models propagation of heat within a given object examples: a heated wire (1D) a metal plate, heated or cooled at its boundaries (2D) water cooling in a reactor (3D problem) air conditioning: where to place ventilations for heating or cooling boiling water in a pot function of interest: temperature T T (x, t), T (x, y), T (x, y, t) or T (x, y, z, t) or... heat propagation depends on the material s heat conductivity. Page 4 of 19

5 3.2. Derivation of the Heat Equation basic idea for modelling: analyse conservation of energy given an arbitrary part D of our computational domain the change of heat energy (per time) is a result of transfer of heat energy across D s surface, heat sources and drains in D (external influences) ρct dv = q dv + k T n ds t D D density ρ, specific heat c, and heat conductivity k are material parameters heat sources and drains are modelled in term q n is outer normal vector on surface element dv/ds according to Gauß theorem: k T n ds = k T dv D D D Page 5 of 19

6 leads to integral equation for arbitrary domain D: ρct t q k T dv = D hence, the integrand has to be identically : T t = κ T + q ρc, κ := k ρc κ > is called the thermal diffusion coefficient (since the Laplace operator models a (heat) diffusion process) Heat Equation: For vanishing external influence q =, we finally get the heat equation: T t = κ T alternate notation T t = κ ( ) 2 T x + 2 T 2 y + 2 T 2 z 2 Page 6 of 19

7 3.3. Boundary and Initial Conditions the heat equation needs boundary or initial-boundary conditions to provide a unique solution Dirichlet boundary conditions: fix T on (part of) the boundary Neumann boundary conditions: T (x, y, z) = ϕ(x, y, z) fix T s normal derivative on (part of) the boundary: T (x, y, z) = ϕ(x, y, z) n pure Dirichlet conditions and mixtures of Dirichlet and Neumann conditions lead to a unique solution. pure Neumann b.c. determines solution up to a constant (if T is a solution, T + c solves the PDE, as well) Page 7 of 19

8 Physical interpretation of boundary conditions: Dirichlet conditions: the temperature T is prescribed itself along (part of) the boundary (some defined heating or cooling) Neumann conditions: the temperature flux across (part of) the boundary is prescribed special case: homogeneous Neumann conditions: complete isolation, no orthogonal transport of heat into or out of the domain For time-dependent heat equation(s): initial conditions required, e.g. for t = Analytical solution: can be given for simple configurations (separation of variables/fourier s method, see exercises) heat equation is one of few examples of PDE, where general statements concerning existence and uniqueness of solutions are possible. Page 8 of 19

9 4. Poisson s Equation in One Dimension We remember the general heat conduction equation, T t = κ T + q ρc. Assuming a steady state problem (T t = ) on the unit interval (i.e. in the 1D case), we get Poisson s equation: or, using an alternate notation: T xx (x) = f(x), x (, 1) u xx (x) = f(x) or even u (x) = f(x) f(x) combines all material parameters and the source term q. Further, we assume homogeneous Dirichlet boundary conditions: u() = u(1) = Page 9 of 19

10 5. Recalling a fundamental theorem of calculus, we may state that x for a suitable constant c 1, and similar u(x) = c 1 + u (ξ) dξ (1) x u (x) = c 2 + u (ζ) dζ (2) Now, suppose that u(x) satisfies the PDE ( u = f(x)), then: u (x) = c 2 f(ζ) dζ ξ Inserting this result into equation (1), we get u(x) = c 1 + c 2 x x ξ f(ζ) dζ dξ Page 1 of 19

11 To simplify the equation, we define ξ F (ξ) := f(ζ) dζ and integrate by parts to get x ξ x x f(ζ) dζ dξ = F (ξ)dξ = [ξf (ξ)] x ξf (ξ)dξ x x x = xf (x) ξf(ξ)dξ = xf(ξ)dξ ξf(ξ)dξ = x (x ξ)f(ξ)dξ Thus, we get as a preliminary result that x u(x) = c 1 + c 2 x (x ξ)f(ξ)dξ (3) Page 11 of 19

12 5.1. Considering Boundary Conditions So far c 1 and c 2 are arbitrary constants. equation (3) solves the PDE in (, 1) the boundary conditions u() = u(1) = have not been taken into account yet. We therefore compute c 1 and c 2 to satisfy u() = u(1) =. u() = leads to u() = c 1 = u(1) = leads to 1 u(x) = c 2 (1 ξ)f(ξ)dξ = or 1 c 2 = (1 ξ)f(ξ)dξ Page 12 of 19

13 5.2. Green s Function As a final result, we get 1 x u(x) = x (1 ξ)f(ξ)dξ (x ξ)f(ξ)dξ being the solution of our boundary value problem. Examples: f(x) = 1, then u(x) = x 1 (1 ξ)dξ x (x ξ)dξ = 1 x(1 x) 2 f(x) = x (see homework) f(x) = e x, then (see Maple worksheet) u(x) = 1 e x + (e 1)x Page 13 of 19

14 Green s Function: Introduce the function G(x, ξ) := { ξ(1 x) if ξ x x(1 ξ) if x ξ 1 then our solution u(x) can be simplified to 1 u(x) = G(x, ξ)f(ξ) dξ (4) Properties: G is continuous G is symmetric: G(x, y) = G(y, x) G(, ξ) = G(1, ξ) = G(x, ) = G(x, 1) = G is a piecewise linear function of x for fixed ξ (and vice versa!) G is positive: G(x, ξ) for all x, ξ [, 1] Page 14 of 19

15 5.3. A Maximum Principle For any continuous function f C([, 1]), let f = sup f(x) x [,1] (maximum norm) Maximum principle (proposition): If u is our solution given by equation (4), then u 1 8 f if f is continuous on [, 1], i.e. f C([, 1]). Proof: 1 1 u(x) G(x, ξ) f(ξ) dξ f G(x, ξ)dξ A short computation leads to Page 15 of 19 u 1 8 f

16 5.4. Uniqueness of the Solution Consider two boundary value problems u xx (x) = f 1 (x) and u xx (x) = f 2 (x), u() = u(1) =, with their respective solutions u 1 (x) and u 2 (x). Due to the linearity of the problem, u 1 (x) u 2 (x) is then a solution of the boundary value problem u xx (x) = f 1 (x) f 2 (x). If f 1 = f 2 this means that, if the solutions u 1 and u 2 were actually different, the maximum principle would require that u 1 u f 1 f 2 = Hence, u 1 and u 2 must be identical. That means, the solution to our 1D boundary value problem is unique! Page 16 of 19

17 6. A Finite Difference Approximation Similar to numerical Methods for ODE, we will approximate the solution on a grid of n equidistant grid points (meshsize h := 1 n ): x i := ih, i =,..., n; compute approximations u i u(x i ) for i =,..., n; replace the second derivative by an appropriate difference term u (x i ) = u i+1 2u i + u i 1 h 2 where the error term E h (x) is O(h 2 ); + E h (x) Thus, we get the following system equations: u i+1 2u i + u i 1 h 2 = f(x i ) for i = 1,..., n 1; (5) with boundary conditions u = u() = and u n = u(1) =. Page 17 of 19

18 6.1. Setting up a Linear System of Equations The linear system of equations (5) can also be written in matrixvector-notation: Au = b where u = (u 1,..., u n 1 ) T is the vector of unknowns; b = (b 1,..., b n 1 ) T, with component b i given by b i := h 2 f(x i ) for i = 1,..., n 1; the system matrix A is given by A = Page 18 of 19

19 6.2. Properties of the Linear System of Equations the matrix A is diagonal dominant, which means that for matrix elements a ij the equation a ii i j a ij holds for every i = 1,..., n 1 as a consequence, all eigenvalues of A are positive, and A is non-singular. therefore, the linear system of equations has a unique solution Page 19 of 19