Math Problem Statement
Solve ∂u ∂t = α 2 ∂ 2u ∂x2 subject to the conditions (i) u is not infinite as t −→ ∞ (ii) u = 0 for x = 0 and x = π for all t (iii) u = πx − x 2 for t = 0 in (0, π)
Solution
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Math Problem Analysis
Mathematical Concepts
Partial Differential Equations
Heat Equation
Separation of Variables
Fourier Series
Formulas
∂u/∂t = α^2 ∂^2u/∂x^2
General solution u(x, t) = Σ C_n sin(nx) e^{-α^2 n^2 t}
Fourier coefficients C_n = (2/π) ∫_0^π (πx - x^2) sin(nx) dx
Theorems
Separation of Variables
Fourier Series Expansion
Suitable Grade Level
University Level (Advanced Calculus/PDEs)
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