Q8
(a) Reduce the following partial differential equation to a canonical form and hence solve it: $$yu_{xx} + (x+y)u_{xy} + xu_{yy} = 0$$ (15 marks) (b) Using Runge-Kutta method of fourth order, solve the differential equation $\frac{dy}{dx} = x + y^2$ with $y(0) = 1$, at $x = 0.2$. Use four decimal places for calculation and step length 0.1. (15 marks) (c) Verify that $w = ik \log \{(z-ia)/(z+ia)\}$ is the complex potential of a steady flow of fluid about a circular cylinder, where the plane $y = 0$ is a rigid boundary. Find also the force exerted by the fluid on unit length of the cylinder. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) निम्नलिखित आंशिक अवकल समीकरण $$yu_{xx} + (x+y)u_{xy} + xu_{yy} = 0$$ को विहित रूप में समानीत कीजिये और अतः इसको हल कीजिये। (15 अंक) (b) चतुर्थ कोटि की रूने-कुट्टा विधि का उपयोग करके अवकल समीकरण $\frac{dy}{dx} = x + y^2$, जबकि $y(0) = 1$ है, को $x = 0.2$ पर हल कीजिये। परिकलन में दशमलव के चार स्थानों तक तथा पग लम्बाई (स्टेप लैंथ) 0.1 का उपयोग कीजिये। (15 अंक) (c) सत्यापित कीजिये कि एक वृत्ताकार बेलन के इर्द-गिर्द एक तरल के अपरिवर्ती प्रवाह का सम्मिश्र विभव $w = ik \log \{(z-ia)/(z+ia)\}$ है, जहाँ समतल $y = 0$ एक दृढ़ सीमा है। बेलन की एकक लम्बाई (यूनिट लैंथ) पर तरल द्वारा लगाये गये बल को भी ज्ञात कीजिये। (20 अंक)
Directive word: Solve
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How this answer will be evaluated
Approach
Solve all three sub-parts systematically, allocating approximately 30% time to part (a) on PDE canonical reduction, 30% to part (b) on RK4 numerical computation, and 40% to part (c) on complex potential verification and force calculation. Begin with identifying the PDE type and characteristic equations for (a), execute precise iterative calculations for (b), and apply Blasius theorem or residue calculus for force in (c). Present each part with clear headings and final boxed answers.
Key points expected
- For (a): Correct classification of PDE type (hyperbolic/parabolic/elliptic) via discriminant B²-4AC = (x+y)²-4xy = (x-y)² ≥ 0, identifying hyperbolic nature with characteristic curves ξ = x-y and η = x+y or similar
- For (a): Proper reduction to canonical form u_ξη = 0 or equivalent, followed by general solution u = f(x-y) + g(x+y) or variant
- For (b): Correct RK4 formulas with k₁, k₂, k₃, k₄ calculations at each step, showing two complete steps (h=0.1) to reach x=0.2 with y(0.2) ≈ 1.2734
- For (c): Verification that w = ik log[(z-ia)/(z+ia)] represents flow past cylinder |z|=a with y=0 as streamline, using conformal mapping properties and image system
- For (c): Application of Blasius theorem or pressure integration to find force, showing zero drag (d'Alembert paradox) and lift calculation using circulation Γ = 2πk
- For (c): Final force expression as (0, 2πρk²) or equivalent per unit length, with proper physical interpretation
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies PDE as hyperbolic with discriminant (x-y)² and finds characteristic curves; for (b): sets up RK4 with proper initial conditions and step size; for (c): correctly interprets complex potential with source/sink at z=±ia and cylinder |z|=a | Identifies PDE type correctly but makes minor errors in characteristic equations; sets up RK4 with correct formula but wrong initial values; recognizes complex potential structure but confuses flow configuration | Misclassifies PDE type or uses wrong discriminant; confuses RK4 with lower-order methods or wrong step size; fails to identify the cylinder boundary or image system in complex potential |
| Method choice | 20% | 10 | For (a): uses standard canonical transformation method with proper Jacobian handling; for (b): applies classical RK4 algorithm with precise coefficient weighting (1:2:2:1); for (c): employs Blasius theorem or residue method for force, with conformal mapping verification | Uses correct general method but with inefficient coordinate choices; applies RK4 with minor coefficient errors; uses pressure integration instead of Blasius theorem or makes residue calculation errors | Attempts wrong method (e.g., separation of variables for (a)); uses Euler or modified Euler instead of RK4; attempts direct integration without complex analysis tools for (c) |
| Computation accuracy | 20% | 10 | For (a): exact algebraic simplification to u_ξη = 0; for (b): precise arithmetic with 4 decimal places yielding y(0.1) ≈ 1.1103 and y(0.2) ≈ 1.2734; for (c): correct residue calculation giving circulation 2πk and force components | Minor algebraic slips in canonical form coefficients; RK4 with one arithmetic error propagating to final answer; correct force direction but wrong magnitude due to factor errors | Major errors in chain rule application for (a); RK4 with wrong formula or complete arithmetic breakdown; nonsensical force values or wrong dimensions in final answer |
| Step justification | 20% | 10 | Explicitly shows: characteristic equation derivation dy/dx = x/y and 1; RK4 intermediate values at each stage; verification that |z|=a is streamline and y=0 is boundary; clear statement of Blasius theorem application with contour choice | Shows key steps but skips some intermediate algebra; presents RK4 calculations without labeling k-values; states results without proving boundary conditions satisfied | Jumps to answers without derivation; presents final numbers only for RK4; asserts force formula without any derivation or theorem reference |
| Final answer & units | 20% | 10 | Clear presentation: (a) general solution u = f(x-y) + g(x+y) with arbitrary functions; (b) y(0.2) = 1.2734 (4 d.p.); (c) force per unit length = 2πρk² in y-direction (lift) with zero drag, proper units [ML⁻¹T⁻²] or dimensionless form stated | Correct functional form for (a) but missing arbitrary functions; correct numerical value for (b) but wrong decimal places; correct force direction but missing factor or units | No final answer for one or more parts; numerical answer without context; completely wrong force expression or missing physical interpretation |
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