1:45 P Mock 1 1 (10 marks) (a) Determine the nature of the PDE -Uxx +...

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1:45 P Mock 1 1 (10 marks) (a) Determine the nature of the PDE -Uxx + xy + y = 0 and bring it to canonical form by applying a suitable variable transformation. (b) Hence obtain the general solution to the PDE of part (a). (c) Using any results you need from lectures, obtain the general solution to the PDE (Hint: Guess a form for the particular solution). 3 (10 marks) Consider the eigenvalue equation 2 (10 marks) Let a > 0 be a constant. Using separation of variables, solve the PDE Urx + y + 2y = 0, 0≤x, y ≤ a with boundary conditions u(0, y) = u(a, y) = u(x,0) = 0, u(r, a) = g(ar). Express the solution u(x, y) as an infinite sum of separable solutions and show how to determine the coefficients. 1/1 -Uvw + W xy + y₂ = y₁ y" + 2x 2² +1²4² + xy = 0, y+Ay=0, y(0) = y(1) = 0. (a) By multiplying by a suitable integrating factor, bring this system to Sturm-Liouville form. (b) Assuming the eigenvalues A are real, show that eigenfunctions corresponding to different eigen- values are orthogonal with respect to a suitable weight function.. (c) Show that the eigenvalues A are non-negative. 4 (11 marks) Solve the following advection equation modelling the steady flow through a pipe, ||| Jau +0,((a+bx)u)=0 where x>0, {ER, u(t.x=0) = f(t). where a > 0, b>0, u = u(t, r) is the concentration of a substance diluted in the fluid and f = f(t) > 0 is the given concentration at x = 0. 5 Huygens principle for solutions to the wave equation VOD) LTE LTE2 Vo)) LTE1 (Ouc² Au=0 where > 0, x ER", u(t=0, x) = g(x), Qu(t = 0.x) h(x), = a) (5 marks) Discuss and formulate d'Alembert's formula in 1D (n = 1) and sketch the cone of dependence for such solutions. How does it reflects Huygens principle? (max 70 words) b) (3 marks) In comparison, write Kirchhoff's integral formula for whole-space solutions to the 3D wave equation (n = 3) and discuss how it reflects Huygens' principle (max 50 words). f(x;p) := 6 (11 marks) Prove Darboux's result, namely that for a given function / C²(R³) the following quantity. 1 Aap Joning f(w)ds (w), satisfies the wave equation Oppf-Af = 0. Handwritten solutions needed O r 1:45 P Mock 1 1 (10 marks) (a) Determine the nature of the PDE -Uxx + xy + y = 0 and bring it to canonical form by applying a suitable variable transformation. (b) Hence obtain the general solution to the PDE of part (a). (c) Using any results you need from lectures, obtain the general solution to the PDE (Hint: Guess a form for the particular solution). 3 (10 marks) Consider the eigenvalue equation 2 (10 marks) Let a > 0 be a constant. Using separation of variables, solve the PDE Urx + y + 2y = 0, 0≤x, y ≤ a with boundary conditions u(0, y) = u(a, y) = u(x,0) = 0, u(r, a) = g(ar). Express the solution u(x, y) as an infinite sum of separable solutions and show how to determine the coefficients. 1/1 -Uvw + W xy + y₂ = y₁ y" + 2x 2² +1²4² + xy = 0, y+Ay=0, y(0) = y(1) = 0. (a) By multiplying by a suitable integrating factor, bring this system to Sturm-Liouville form. (b) Assuming the eigenvalues A are real, show that eigenfunctions corresponding to different eigen- values are orthogonal with respect to a suitable weight function.. (c) Show that the eigenvalues A are non-negative. 4 (11 marks) Solve the following advection equation modelling the steady flow through a pipe, ||| Jau +0,((a+bx)u)=0 where x>0, {ER, u(t.x=0) = f(t). where a > 0, b>0, u = u(t, r) is the concentration of a substance diluted in the fluid and f = f(t) > 0 is the given concentration at x = 0. 5 Huygens principle for solutions to the wave equation VOD) LTE LTE2 Vo)) LTE1 (Ouc² Au=0 where > 0, x ER", u(t=0, x) = g(x), Qu(t = 0.x) h(x), = a) (5 marks) Discuss and formulate d'Alembert's formula in 1D (n = 1) and sketch the cone of dependence for such solutions. How does it reflects Huygens principle? (max 70 words) b) (3 marks) In comparison, write Kirchhoff's integral formula for whole-space solutions to the 3D wave equation (n = 3) and discuss how it reflects Huygens' principle (max 50 words). f(x;p) := 6 (11 marks) Prove Darboux's result, namely that for a given function / C²(R³) the following quantity. 1 Aap Joning f(w)ds (w), satisfies the wave equation Oppf-Af = 0. Handwritten solutions needed O r

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Question 1 a The given PDE is a quasilinear second order PDE of the form uu fxyu 0 To bring this PDE to canonical form we must apply a suitable variable transformation To do this let us define a new v... View the full answer