PhD Preliminary Qualifying Examination: Differential equations (6410/20)

Consider the linear non–autonomous first order system \[ \dot {x}=Ax+B(t)x,\qquad x\in \mathbb {R}^n \] with \(A\) non-singular and \(B(t)\) continuous for \(t\ge 0\). Further, assume that

• the eigenvalues of \(A\) have non–positive real parts and those with zero real part are non-degenerate
• \(\displaystyle \int _0^\infty \lVert B(t)\rVert \dd t=c_1\) with \(c_1\) a positive constant.

1. Let \(\Phi (t)\) be the fundamental matrix of the equation \(\dot {x}=Ax\) with \(\Phi (0)=I\). Derive the variation of constants formula \[ x(t)=\Phi (t)x(0)+\int _0^t \Phi (t-s)B(s)x(s)\dd s \]
2. Prove that the solution \(x(t)\) is bounded for all times \(t>0\). [Hint: Use part (a) and Gronwall’s lemma in the following form: \(v(t)\le v_0+\int _0^t u(s)v(s)\dd s\) implies that \(v(t)\le v_0\exp (\int _0^t u(s)\dd s)\) for \(t>0\)]. What does this imply about the stability of the origin?

Consider the \(T\)-periodic non-autonomous linear differential equation \[ \dot {x}=A(t)x,\qquad x\in \mathbb {R}^n,\qquad A(t)=A(t+T) \] Let \(\Phi (t)\) be a fundamental matrix with \(\Phi (0)=I\).

1. Show that there exists at least one nontrivial solution \(\chi (t)\) such that \[ \chi (t+T)=\mu \chi (t) \] where \(\mu \) is an eigenvalue of \(\Phi (T)\).
2. Suppose that \(\Phi (T)\) has \(n\) distinct eigenvalues \(\mu _i\), \(i=1,\ldots ,n\). Show that there are then \(n\) linearly independent solutions of the form \[ x_i=p_i(t)e^{\rho _i t} \] where the \(p_i(t)\) are \(T\)-periodic. How is \(\rho _i\) related to \(\mu _i\)?
3. Consider the equation \(\dot {x}=f(t)A_0x\), \(x\in \mathbb {R}^2\), with \(f(t)\) a scalar \(T\)-periodic function and \(A_0\) a constant matrix with real distinct eigenvalues. Determine the corresponding Floquet multipliers.

Consider the scalar equation \[ \ddot {x}+\dot {x}=-\epsilon (x^2-x),\qquad 0<\epsilon \ll 1 \] Using the method of multiple scales show that the \(O(1)\) solution is \[ x_0(t,\tau )=A(\tau )+B(\tau )e^{-t}, \] where \(\tau =\epsilon t\), and identify any resonant terms at \(O(\epsilon )\). Show that the non–resonance condition for the amplitude \(A\) is \[ A_\tau =A-A^2 \] and hence determine the asymptotic behavior of \(x_0\). Comment on the domain of validity of the asymptotic expansion.

Consider the scalar differential equation \[ \ddot {x}+x=-\epsilon f(x,\dot {x}) \] with \(\lvert \epsilon \rvert \ll 1\). Let \(y=\dot {x}\).

1. Show that if \(E(x,y)=(x^2+y^2)/2\) then \[ \dot {E}=-\epsilon f(x,y)y. \] Hence show that an approximate periodic solution of the form \(x=A\cos t+O(\epsilon )\) exists if \[ \int _0^{2\pi } f(A\cos t,-A\sin t)\sin td t=0. \]
2. Let \(E_n=E(x(2\pi n),y(2\pi n))\) with \(x(t)=A_n\cos t+O(\epsilon )\) for \(2\pi n\le t<2\pi (n+1)\). Show that to lowest order \(E_n\) satisfies a difference equation of the form \[ E_{n+1}=E_n+\epsilon F(E_n) \] and write down \(F(E_n)\) explicitly as an integral. Hence deduce that a periodic orbit with approximate amplitude \(A^*=\sqrt {2E^*}\) exists if \(F(E^*)=0\) and this orbit is stable if \[ \epsilon \frac {dF}{dE}(E^*)<0. \]
3. Using the above result, find the approximate amplitude of the periodic orbit of the van der Pol equation \[ \ddot {x}+x+\epsilon (x^2-1)\dot {x}=0 \] and verify that it is stable.

The displacement \(x\) of a spring–mounted mass under the action of dry friction is assumed to satisfy \[ \ddot {x}+x=F_0\operatorname {sgn}(v_0-\dot {x}) \]

1. Calculate the phase paths in the \((x,y)\) plane and draw the phase diagram. Deduce that the system ultimately converges into a limit cycle oscillation. What happens when \(v_0=0\)?
2. Suppose \(v_0=0\) and the initial conditons at \(t=0\) are \(x=x_0>3F_0\) and \(\dot {x}=0\). Subsequently, whenever \(x=-\alpha \), where \(2F_0-x_0<-\alpha <0\) and \(\dot {x}>0\), a trigger operates to increase suddenly the forward velocity so that the kinetic energy increases by a constant amount \(E\). Show that if \(E>8F_0^2\) then a periodic motion is approached, and show that the largest value of \(x\) in the periodic motion is equal to \(F_0+E/(4F_0)\).

Consider the scalar linear equation \[ a\frac {\partial u}{\partial x}+b\frac {\partial u}{\partial y}=\alpha u \] with \(a,b,\alpha \) differentiable functions of \(x,y\). Suppose that \(u\) is prescribed on some arc \(\Gamma \) (Cauchy data).

1. By introducing a test function \(\psi \) that vanishes on an arbitrary curve \(\gamma \) and applying Green’s theorem to the domain \(D\) bounded by \(\partial D=\Gamma \cup \gamma \), derive the integral equation for a weak solution to the Cauchy problem.
2. Extend the analysis to the case where \(u\) is discontinuous across an open curve \(C_0\) within the domain \(D\). Hence show that \(C_0\) is a characteristic projection.
3. Write down the generalization of part (b) to the case of the quaslinear equation \[ \frac {\partial P}{\partial x}+\frac {\partial Q}{\partial y}=c \] where \(P,Q,c\) are differentiable functions of \(x,y,u\), and hence derive the Rankine–Hugonoit condition.

Paint flowing down a wall has thickness \(u(x,t)\) satisfying (for \(t>0\)) \[ \frac {\partial u}{\partial t}+u^2\frac {\partial u}{\partial x}=0. \]

1. Show that the characteristics are straight lines and that the Rankine–Hugoniot condition on a shock \(x=S(t)\) is \[ \frac {dS}{dt}=\frac {\left [u^3/3\right ]_-^+}{[u]_-^+}. \]
2. A stripe of paint is applied at \(t=0\) so that \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1\\ 1, & 0<x<1 \end {cases} \] For sufficiently small \(t\), determine \(u\) in the domains \(x<0\), \(0<x<t\), \(t<x<S(t)\) and \(S(t)<x\), where the shock is \(x=S(t)=1+t/3\)
3. Explain why the solution changes at \(t=3/2\) and show that thereafter \(\dot {S}=S/3t\).

(a) Consider the three–dimensional wave equation \[ \frac {\partial ^2u}{\partial t^2}=c^2\nabla ^2u \] with Cauchy data \[ u(\mathbf {x},0)=0,\qquad \frac {\partial u}{\partial t}(\mathbf {x},0)=g(\mathbf {x}) \]

1. Show that radially symmetric solutions are in the form of outgoing and incoming waves \[ u(r,t)=\frac {1}{r}F(r\pm ct). \] [Hint: Perform the substitution \(v=ur\).]
2. Writing the general solution to the Cauchy problem as a superposition of outgoing waves \[ u(\mathbf {x},t)=\int _{\mathbb {R}^3}\frac {\delta (r-ct)}{r}f(x',y',z')\dd x'\dd y'\dd z', \] where \(r^2=(x-x')^2+(y-y')^2+(z-z')^2\), derive the retarded potential solution \[ \begin {aligned} u(\mathbf {x},t)=ct\int _0^{2\pi }\int _0^\pi &f(x+ct\sin \theta \cos \phi ,\ y+ct\sin \theta \sin \phi ,\\ &z+ct\cos \theta )\sin \theta \dd \theta \dd \phi \end {aligned} \] and show that \(f=g/(4\pi c)\).
3. Assuming that the initial data \(g\) is only nonzero in a bounded domain \(D\), use a graphical construction to derive Huygen’s principle.

1. Show that if the real symmetric matrix \(A\) has real eigenvalues \(\lambda _i\) and orthogonal eigenvectors \(\mathbf {x}_i\), then for any vector \(\mathbf {y}=\sum _i c_i\mathbf {x}_i\), the smallest eigenvalue satisfies \[ \lambda _0\le \frac {\mathbf {y}^TA\mathbf {y}}{\mathbf {y}^T\mathbf {y}}. \]
2. Show that the eigenfunctions \(\phi \) and eigenvalues \(-\lambda \) of the problem \[ \nabla ^2\phi +\lambda \phi =0\quad \text {in a region }D, \] with \[ \frac {\partial \phi }{\partial n}+\alpha \phi =0\quad \text {on }\partial D, \] where \(\partial /\partial n\) is the outward normal derivative, satisfy \[ \lambda \int _D \phi ^2d x =\int _D \lvert \nabla \phi \rvert ^2d x+\alpha \int _{\partial D}\phi ^2d s. \] Assuming that the eigenfunctions \(\phi \) form a complete orthonormal set, derive an upper bound for the smallest eigenvalue in terms of an appropriate energy integral (Rayleigh quotient).

1. Solve Poisson’s equation \[ \nabla ^2u=f\text { in }D,\qquad u=g\text { on }\partial D \] in terms of an appropriately defined Green’s function.
2. Show that the two–dimensional Green’s function for the Laplacian in an unbounded domain is \[ G(\mathbf {x},\mathbf {x}')=\frac {1}{2\pi }\ln \lvert \mathbf {x}-\mathbf {x}'\rvert . \]
3. Use the method of images to derive the Green’s function for the Laplacian in the half–space \(x\in \mathbb {R}\), \(y>0\) with a Dirichlet boundary condition on \(y=0\), and evaluate the corresponding solution of part (a) when \(f=0\).