Assignment Solutions of Partial Difierential Equations

Assignment Solutions of Partial Differential EquationsWeijiu LiuDepartment of MathematicsUniversity of Central Arkansas201 Donaghey Avenue, Conway, AR 72035, USA1 Assignment Derive the heat equation for a rod assuming constant thermal properties with variablecross-sectional areaA(x) assuming no byAthe the cross-sectional quantities: Thermal energy densitye(x, t) = the amount of thermal energy per unit volume. Heat flux (x, t) = the amount of thermal energy flowing across boundaries per unitsurface area per unit time. Heat sourcesQ(x, t) = 0. Temperatureu(x, t). Specific heatc= the heat energy that must be supplied to a unit mass of a substanceto raise its temperature one unit. Mass density (x) = mass per unit volume. Fourier s Law: the heat flux is proportional to the temperature gradient = K0 u.(1)Conservation of heat energy:Rate of change of heat energy in time = Heat energy flowing across boundaries per unittime + Heat energy generated insider per unit time heat energy =e(x, t)A(x) x.

c1 = RL 0 f(x)dx¡ 7 2 L2 ¡ 5 24 L4 L; and then ` = ¡ 1 6 x3 + µ 7+ 1 2 L2 ¶ x+ RL 0 f(x)dx¡ 7 2 L2 ¡ 5 24 L4 L: 1.5.2. For conduction of thermal energy, the heat °ux vector is ` = ¡K0ru: If in addition the molecules move at an average velocity V, a process called convection, then ` = ¡K0ru+c‰uV: Derive the corresponding equation for ...

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