The charge supplied by source varies with time $'t'$ as $Q = at -bt^2$. The total heat produced in the resistor $2R$ before current reduces to zero
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$I=a-2 b t$
current in $2 \mathrm{\,R}$ resistor $=\frac{1}{3}(\mathrm{a}-2 \mathrm{\,bt})$
$H = \int_0^{a/2b} {\frac{1}{3}} {(a - 2\,bt)^2}2R\,dt = \frac{{{a^3}R}}{{27b}}$
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