Differential Equations I Flashcards
Define a Lipschitz function
A function f : R2 → R, satisfies a Lipschitz condition (with Lipschitz constant L) if there exists L > 0 such that |f(x, y1) − f(x, y2)| ≤ L|y1 − y2| for all x ∈ R and all y1, y2 ∈ R.
State Picard’s Theorem
Let f : R → R be a function defined on the rectangle R := {(x, y) : |x − a| ≤ h, |y − b| ≤ k} which satisfies:
P(i): f is continuous in R with |f(x, y)| ≤ M for all (x, y) ∈ R, where Mh ≤ k
P(ii): f satisfies a Lipschitz condition in R.
Then the IVP y’(x) = f(x, y(x)) with y(a) = b. has a unique solution y on the interval [a-h, a+h].
State Gronwall’s Inequality
Suppose A,B ≥ 0 are constants and v is a non-negative continuous function satisfying v(x) ≤ B + A|integral from x to a of v(s)| for all x in interval [a − h1, a + h2], h1,h2 ≥ 0. Then v(x) ≤ Be^(A|x−a|) for all x ∈ [a − h1, a + h2].
State the comparison principle
Let g: R2 → R be continuously differentiable and let u(x) and v(x) be differentiable functions which satisfy u’(x) ≤ g(x, u(x)) and v’(x) ≥ g(x, v(x)) on some interval I. If u(x0) ≤ v(x0) for some x0 ∈ I then we obtain that u(x) ≤ v(x) for all x ∈ I with x ≥ x0 while if u(x0) ≥ v(x0) then we obtain that u(x) ≥ v(x) for all x ∈ I with x ≤ x0.
Define a plane autonomous system
A plane autonomous system of ODEs is a pair of ODEs in the form: dx/dt = X(x,y) and dy/dt = Y(x,y)
Define a critical point of a plane autonomous system
A point (x0, y0) in the phase plane where X(x0, y0) = Y (x0, y0) = 0, so is a point where the solution is constant for all time.
Define a stable critical point
A critical point (a, b) is stable if for all ϵ > 0 there exists δ > 0 and t0 such that sqrt[(x(t0) − a)^2 + (y(t0) − b)^2] < δ implies that sqrt[(x(t) − a)^2 + (y(t) − b)^2] < ϵ, ∀t > t0.
Define an unstable critical point
A critical point (a, b) is unstable if there exists ϵ > 0 such that for all δ > 0 and t0 where sqrt[(x(t0) − a)^2 + (y(t0) − b)^2] < δ, also sqrt[(x(t) − a)^2 + (y(t) − b)^2] ≥ ϵ, ∀t > t0.
Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where 0 < λ1 < λ2, classify the critical point.
Unstable node, with all curves except that on Z1 tending to Z2.
Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1 < λ2 < 0, classify the critical point.
Stable node, with all curves except that on Z2 coming from the Z1 direction.
Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1 = λ2 = λ, classify the critical point.
If M-λI = 0, we can write Z(t) = Ce^(λt) and it is a star, otherwise it is an inflected node. In both cases, if λ > 0, it is unstable and if λ < 0, it is stable.
Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1 < 0 < λ2, classify the critical point.
Saddle.
If c1 = 0, Z(t) → 0,∞ along Z2.
If c2 = 0, Z(t) → 0,∞ along Z1.
Otherwise, Z(t) → 0 along Z1 and Z(t) → ∞ along Z2.
Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1, λ2 are complex are their real parts are 0, classify the critical point.
Centre (periodic)
If the second entry of M is positive, it is clockwise, if it is negative, it is anticlockwise
Given Z(t) = (c1)(Z1)e^(λ1)t + (c2)(Z2)e^(λ2)t, where λ1, λ2 are complex are their real parts are non-zero, classify the critical point.
Spiral
If real part is positive, unstable
If real part is negative, stable
If the second entry of M is positive, it is clockwise, if it is negative, it is anticlockwise
State Bendixson-Dulac’s Theorom
Consider the system dx/dt = X(x,y) and dy/dt = Y(x,y), with X, Y ∈ C1. If there exists a function φ(x, y) ∈ C1 with
ρ = ∂/∂x(φX) + ∂/∂y(φY ) > 0 in a simply connected region R then there can be no nontrivial closed trajectories lying entirely in R.
Give the general form of a linear first order PDE
P(x, y)∂z/∂x + Q(x, y)∂z/∂y = R(x, y, z(x, y)), for some function z.
Define a semi-linear PDE
A PDE is said to be semi-linear if it is linear in the highest order partial derivatives and the coefficients of the highest order partial derivatives depend only on x and y.
Define a characteristic curve of a semi-linear PDE
Given the PDE P(x, y)∂z/∂x + Q(x, y)∂z/∂y = R(x, y, z(x, y)), the curve Γ = (x(t), y(t), z(t)), where x(dot) = P(x, y), y(dot) = Q(x, y) and z(dot) = R(x, y, z) is the characteristic curve.
Define a characteristic projection
Given a characteristic (x(t), y(t), z(t)), the characteristic projection is the curve (x(t), y(t))
Define Cauchy data
If P(x, y)y_s − Q(x, y)x_s ≠ 0 on the data curve, we say the data is Cauchy.
Define the domain of definition
The region of the (x,y) plane where the solution is determined uniquely by the data.
Give the general form of a second order semi-linear PDE
a(x,y)u_xx + 2b(x,y)u_xy + c(x,y)u_yy = f(x,y,u,u_x,u_y)
Given a(x,y)u_xx + 2b(x,y)u_xy + c(x,y)u_yy = f(x,y,u,u_x,u_y), classify the PDE
If ac < b^2: hyperbolic
If ac > b^2: elliptic
If ac = b^2: parabolic
Give the normal form of a hyperbolic equation
uφψ = G(φ, ψ, u, uφ, uψ).
Give the normal form of an elliptic equation
uζζ + uηη = H(ζ, η, u, uζ , uη)
Give the normal form of a parabolic equation
uψψ = G(u, φ, ψ, uφ, uψ).
State the maximum principle for the Laplacian
Suppose u satisfies uxx + uyy ≥ 0, for all (x, y) in some domain D. Then u attains its maximum value on ∂D.
State the maximum principle for the heat equation
Suppose that u satisfies ut − uxx ≤ 0 in a region D bounded by the lines t = 0, t = τ > 0, and two non-intersecting smooth curves C1 and C2 that are nowhere parallel to the x-axis. Then u takes its maximum value either on t = 0 or on one of the curves C1 or C2.
