Q8
(a) Find a complete integral of the partial differential equation p = (z + qy)² by using Charpit's method. 15 marks (b) Derive Newton's backward difference interpolation formula and also do error analysis. 15 marks (c) Show that for the complex potential tan⁻¹z, the streamlines and equipotential curves are circles. Find the velocity at any point and check the singularities at z = ±i. 20 marks
हिंदी में प्रश्न पढ़ें
(a) चारपिट विधि का उपयोग करके आंशिक अवकल समीकरण p = (z + qy)² का पूर्ण समाकल प्राप्त कीजिए। 15 अंक (b) न्यूटन के पश्चातर अंतर्वेशन सूत्र की व्युत्पत्ति कीजिए तथा त्रुटि-विश्लेषण भी कीजिए। 15 अंक (c) दर्शाइए कि सम्मिश्र विभव tan⁻¹z के लिए धारा-रेखाएँ तथा समविभव वक्र, वृत्त हैं। किसी भी बिंदु पर वेग निकालिए तथा z = ±i पर विचित्रता जाँचिए। 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) on Charpit's method (15 marks), 30% to part (b) on Newton's backward interpolation with error analysis (15 marks), and 40% to part (c) on complex potential analysis (20 marks). Begin each part with clear statement of the method being used, show complete derivation steps, and conclude with boxed final answers for each sub-part.
Key points expected
- For (a): Correct formulation of Charpit's auxiliary equations and identification of suitable parameter to integrate, leading to complete integral with two arbitrary constants
- For (a): Proper handling of the non-linear PDE by choosing appropriate differentials and solving the resulting compatible system
- For (b): Complete derivation of Newton's backward difference formula using backward difference operator ∇ and binomial expansion
- For (b): Error analysis showing truncation error term involving ∇ⁿ⁺¹f or f⁽ⁿ⁺¹⁾(ξ), with clear explanation of error order
- For (c): Separation of complex potential tan⁻¹z into real and imaginary parts φ(x,y) and ψ(x,y), showing both satisfy Laplace's equation
- For (c): Proof that streamlines ψ = constant and equipotentials φ = constant form orthogonal families of circles with centers on imaginary and real axes respectively
- For (c): Calculation of velocity components from dw/dz and verification of singularities at z = ±i as simple poles with residue analysis
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): Correctly writes Charpit's equations dx/fp = dy/fq = dz/(pfp+qfq) = -dp/(fx+pfp) = -dq/(fy+qfq) with f = p - (z+qy)²; for (b): Properly defines backward differences ∇f, ∇²f, etc. and sets up interpolation at x = xn + ph; for (c): Correctly identifies w = tan⁻¹z = φ + iψ and uses w = ½i ln[(i-z)/(i+z)] for separation | Writes basic equations for each part but misses subtle setup details like proper parameter choice in (a) or standard form in (b); partial setup for complex potential | Incorrect setup of Charpit's equations, confuses forward/backward differences, or fails to identify real/imaginary parts of complex potential |
| Method choice | 20% | 10 | For (a): Strategic choice of additive separation or parameter method leading to integrable combinations; for (b): Clear choice of backward differences suited for end-point interpolation; for (c): Elegant use of conformal mapping properties and logarithmic form of arctangent | Uses standard methods without optimization; acceptable but lengthy approaches; misses elegant substitutions | Wrong method selection (e.g., Lagrange multipliers instead of Charpit, forward differences for backward formula); no recognition of conformal mapping in (c) |
| Computation accuracy | 20% | 10 | For (a): Flawless integration yielding complete integral z = (y+a)²/(4(x+b)) - a or equivalent with verified arbitrary constants; for (b): Correct binomial expansion coefficients and error term (p+n choose 2n+1)∇ⁿ⁺¹f or hⁿ⁺¹f⁽ⁿ⁺¹⁾(ξ)/(n+1)!; for (c): Accurate velocity dw/dz = 1/(1+z²) and correct residue calculations at ±i | Minor algebraic slips in integration or expansion coefficients; correct final forms with small computational errors; partial velocity calculation | Major integration errors, wrong binomial coefficients, incorrect velocity formula, or failure to identify simple poles at z = ±i |
| Step justification | 20% | 10 | For (a): Justifies choice of parameter and shows compatibility condition; for (b): Proves equivalence of backward difference formula to Newton's form and derives error from Taylor remainder; for (c): Proves orthogonality of circles via Cauchy-Riemann equations and justifies singularity classification using limit behavior | Shows key steps with gaps in rigorous justification; mentions theorems without full application; partial proofs | Unjustified leaps between steps, missing crucial derivations, or assertion without proof for critical claims |
| Final answer & units | 20% | 10 | For (a): Complete integral clearly stated as F(x,y,z,a,b) = 0 with two independent arbitrary constants; for (b): Formula stated in standard notation with explicit error term; for (c): Streamlines x² + (y+c)² = c²-1 and equipotentials (x-c)² + y² = c²-1 explicitly derived, velocity V = 1/|1+z²| with direction, singularities confirmed as simple poles with residues ∓1/2i | Correct final forms but poorly presented; missing some constants or error terms; incomplete description of curves | Missing final answers, wrong forms, or failure to specify nature of singularities and velocity characteristics |
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