Q7
(a) Find the general solution of the partial differential equation (D² - D'² - 3D + 3D')z = xy + e^(x+2y) where D ≡ ∂/∂x and D' ≡ ∂/∂y. 15 marks (b) Solve the system of equations 3x₁ + 9x₂ - 2x₃ = 11 4x₁ + 2x₂ + 13x₃ = 24 4x₁ - 2x₂ + x₃ = -8 correct up to 4 significant figures by using Gauss-Seidel method after verifying whether the method is applicable in your transformed form of the system. 15 marks (c) Show that q⃗ = λ(-yî + xĵ)/(x² + y²), (λ = constant) is a possible incompressible fluid motion. Determine the streamlines. Is the kind of the motion potential? If yes, then find the velocity potential. 20 marks
हिंदी में प्रश्न पढ़ें
(a) आंशिक अवकल समीकरण (D² - D'² - 3D + 3D')z = xy + e^(x+2y) का व्यापक हल प्राप्त कीजिए, जहाँ D ≡ ∂/∂x तथा D' ≡ ∂/∂y है। 15 अंक (b) समीकरणों के निकाय 3x₁ + 9x₂ - 2x₃ = 11 4x₁ + 2x₂ + 13x₃ = 24 4x₁ - 2x₂ + x₃ = -8 का गॉस-सीडल विधि द्वारा 4 सार्थक अंकों तक सही हल प्राप्त कीजिए, यह सत्यापन करने के बाद कि क्या यह विधि आपके द्वारा निकाय के रूपांतरित रूप में अनुप्रयोज्य है। 15 अंक (c) दर्शाइए कि q⃗ = λ(-yî + xĵ)/(x² + y²), (λ = स्थिरांक) एक संभाव्य असंपीड्य तरल गति है। धारा-रेखाएँ निकालिए। क्या गति का प्रकार विभव है? यदि हाँ, तो वेग विभव निकालिए। 20 अंक
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How this answer will be evaluated
Approach
Solve this multi-part numerical problem by allocating approximately 30% time to part (a) PDE solution, 30% to part (b) Gauss-Seidel iterative method with convergence verification, and 40% to part (c) fluid dynamics analysis including streamline equations and velocity potential determination. Begin with clear problem identification for each part, show complete working with proper mathematical notation, and conclude with verified final answers for all three components.
Key points expected
- Part (a): Factorize the operator (D² - D'² - 3D + 3D') as (D-D')(D+D'-3), find complementary function through characteristic equations, and determine particular integral for both xy and e^(x+2y) terms using appropriate methods
- Part (b): Verify diagonal dominance or transform the system to ensure convergence, rearrange equations if needed, apply Gauss-Seidel iteration formula with correct update sequence, and iterate until 4 significant figure accuracy is achieved
- Part (c): Verify incompressibility condition ∇·q⃗ = 0, derive streamline equations dy/dx = -x/y leading to x² + y² = constant, check irrotationality (∇×q⃗ = 0) to determine if motion is potential, and find velocity potential φ = -λ tan⁻¹(y/x) or equivalent
- Correct handling of non-homogeneous terms in PDE: polynomial and exponential particular integrals with proper operator substitution
- Gauss-Seidel convergence criterion: strict diagonal dominance or symmetric positive definite matrix verification before iteration
- Streamline and velocity potential relationship: demonstration that φ exists only when flow is irrotational, with explicit calculation of vorticity
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly factorizes PDE operator in (a), verifies convergence conditions (diagonal dominance or matrix properties) before applying Gauss-Seidel in (b), and properly sets up divergence and curl calculations for fluid verification in (c) | Partially correct setup with minor errors in operator factorization or convergence verification, or incomplete fluid dynamics setup | Incorrect operator factorization, failure to verify convergence before iteration, or missing incompressibility/irrotationality setup |
| Method choice | 20% | 10 | Selects appropriate method for particular integral (trial function for xy, shift theorem for exponential), chooses correct iterative scheme with proper equation ordering, and applies correct tests for potential flow | Correct general approach with some inefficiency or minor method errors, such as using Jacobi instead of Gauss-Seidel or suboptimal PI method | Inappropriate method selection such as wrong PI technique, using direct methods instead of iteration for (b), or incorrect potential flow test |
| Computation accuracy | 20% | 10 | Accurate algebraic manipulation throughout, correct iteration values converging to 4 significant figures (typically x₁≈-1.000, x₂≈2.000, x₃≈2.000), precise streamline integration and correct velocity potential derivation | Minor computational errors in coefficients or iteration values that don't converge properly, or sign errors in streamline equations | Major computational errors leading to wrong complementary function, divergent iterations, or incorrect velocity potential by factor of λ or wrong functional form |
| Step justification | 20% | 10 | Clear justification for operator factorization, explicit convergence verification with matrix transformation details if needed, and rigorous proof of irrotationality with vorticity calculation; all steps logically connected | Some steps justified but gaps in reasoning, such as stating convergence without proof or asserting potential flow without curl calculation | Missing critical justifications: no convergence check, unexplained operator factorization, or assertion of potential flow without verification |
| Final answer & units | 20% | 10 | Complete general solution for (a) with CF and PI clearly stated, converged solution to 4 significant figures for (b), and complete streamline equations x²+y²=c with velocity potential φ=-λθ (or tan⁻¹ form) for (c); all parts clearly labeled | Correct final answers but poorly formatted, missing some components, or incorrect significant figures in iteration results | Incomplete or missing final answers, wrong form of solution (e.g., missing arbitrary functions in PDE), or no clear statement of velocity potential existence |
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