The slab with accumulation of a radiogenic isotope

In K-Ar or zircon U-Pb dating, modeling the loss of radiogenic isotopes by volume diffusion is important. If P0 is the local concentration at t = 0 of a radioactive element decaying with constant X, a source term exists in the transport equation of the radiogenic element which is the local rate of accumulation AP0e Xt. For dual decay, 

At t=0, the slab is supposed to be free of radiogenic element (C0=0). At the boundaries x=0 and x = X of the system, concentration is kept to zero. For simplicity, P0 will be assumed to be constant over the mineral. 

Let us consider in the first place the total concentration N = C + P0e M of radioactive and radiogenic isotopes. Since there is no loss of the radioactive isotope, the variation of N equals the loss of the radiogenic isotope. In other words 

Alternatively, the solution can be worked out through a series of steps similar to those taken for the non-radioactive case. We assume that a solution can be found as a product of a function fit) and a series of trigonometric functions in x such as 

The trouble is now that the source term does not include the sum of sines, so we will use a trick resting on the Leibniz s rule for differentiating integrals. A particular solution of the diffusion equation with radiogenic accumulation is 

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