Solution of the boundary value of the problem \(\frac{{{\partial ^2}T}}{{\partial t}} = \frac{{{\partial ^2}T}}{{\partial {x^2}}},\)subject to the conditions1. T = 0 at t → ∞,2. T = 0 for x = 0 and3. x = a for all t > 0, T = x when t = 0 and 0 < x < a

• A. \(T\left( {x,\;t} \right) = \frac{{2a}}{\pi }\mathop \sum \limits_{n = 1}^\infty {\left( { - 1} \right)^{\frac{{n - 1}}{n}}}\sin \frac{{n\pi }}{x}\exp \left[ { - {{\left( {\frac{{n\pi }}{a}} \right)}^2}t} \right]\)
• B. \(T\left( {x,t} \right) = \mathop \sum \limits_l^\infty \frac{{\cos m\pi ct}}{t}\frac{{\sin m\pi x}}{t}\)
• C. \(T\left( {x,t} \right) = \mathop \sum \limits_l^\infty \frac{{\cos m\pi t}}{t}\frac{{\sin m\pi cx}}{t}\)
• D. \(T\left( {x,t} \right)= \mathop \sum \limits_l^\infty {\left( { - 1} \right)^{\frac{1}{n}}}\frac{{\cos m\pi ct}}{t}\exp \left[ { - {{\left( {\frac{{n\pi }}{a}} \right)}^2}t} \right]\)