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$$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$
with an initial condition and boundary conditions
$$T(x,0)=T_0$$
$$T(L,t)=T_0$$
$$k\left.\frac{\partial T}{\partial x}\right_{x=0}=2A\cos^2\left(\frac{\omega t}{2}\right)=A(\cos\omega t+1)$$
where $A=V_0^2/(8RhL)$, $V_0$ is the voltage applied to the heater, R the electrical resistance of the heater, h the thickness of the thin film, $\alpha$ the thermal diffusivity of the thin film, and $\omega/2$ the heating frequency. The solution for this problem is
$$T(x,t)T_0=\frac Ak\sqrt{\frac\alpha\omega}\exp\left (\sqrt{\frac{\omega}{2\alpha}}x\right)\\ \times\cos\left(\omega t\sqrt{\frac{\omega}{2\alpha}}x\frac\pi4\right)\frac Ak(xL)$$
I got this from a paper. I'm trying to derive how the author came to the solution from the boundary conditions. There is no derivation in the paper, and I searched books and the internet thoroughly but couldn't find anything. I attached the solution I'm getting. But I don't think it's correct! Any help regarding where I'm doing wrong will be greatly appreciated.
with an initial condition and boundary conditions
$$T(x,0)=T_0$$
$$T(L,t)=T_0$$
$$k\left.\frac{\partial T}{\partial x}\right_{x=0}=2A\cos^2\left(\frac{\omega t}{2}\right)=A(\cos\omega t+1)$$
where $A=V_0^2/(8RhL)$, $V_0$ is the voltage applied to the heater, R the electrical resistance of the heater, h the thickness of the thin film, $\alpha$ the thermal diffusivity of the thin film, and $\omega/2$ the heating frequency. The solution for this problem is
$$T(x,t)T_0=\frac Ak\sqrt{\frac\alpha\omega}\exp\left (\sqrt{\frac{\omega}{2\alpha}}x\right)\\ \times\cos\left(\omega t\sqrt{\frac{\omega}{2\alpha}}x\frac\pi4\right)\frac Ak(xL)$$
I got this from a paper. I'm trying to derive how the author came to the solution from the boundary conditions. There is no derivation in the paper, and I searched books and the internet thoroughly but couldn't find anything. I attached the solution I'm getting. But I don't think it's correct! Any help regarding where I'm doing wrong will be greatly appreciated.
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