The advection-diffusion equation was $\frac{\partial c}{\partial t} = \nabla \cdot (D\nabla c) - \nabla \cdot (vc) + R$ where $c$ is a scalar (the concentration) and $v=(v_1 ,v_2 ,v_3 )$ is a vector (the velocity). I assumed everything could be a function of the three space coordinates $x, y, z$ and the time $t$. Then the equation becomes
$ \begin{align*} \frac{\partial c}{\partial t} & = D\left(\frac{\partial^2 c}{\partial x^2} + \frac{\partial^2 c}{\partial y^2} + \frac{\partial^2 c}{\partial z^2}\right) + \frac{\partial D}{\partial x}\frac{\partial c}{\partial x} + \frac{\partial D}{\partial y}\frac{\partial c}{\partial y} + \frac{\partial D}{\partial z}\frac{\partial c}{\partial z} - \frac{\partial (cv_1)}{\partial x} - \frac{\partial (cv_2)}{\partial y} - \frac{\partial (cv_3)}{\partial z} + R\\ & = D\left(\frac{\partial^2 c}{\partial x^2} + \frac{\partial^2 c}{\partial y^2} + \frac{\partial^2 c}{\partial z^2}\right) + \left(\frac{\partial D}{\partial x} - v_1\right)\frac{\partial c}{\partial x} + \left(\frac{\partial D}{\partial y} - v_2\right)\frac{\partial c}{\partial y} + \left(\frac{\partial D}{\partial z} - v_3\right)\frac{\partial c}{\partial z} - c\left(\frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z}\right) + R \end{align*} $
I don't know what else you want. Of course if e.g. $D$ is constant you can set the partial derivatives of $D$ to $0$ , or if the flow is incompressible you can set $\frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z}$ to $0$ . But no such assumptions were made in the original question.
My university advisor has told me that the problem may be solved as a one dimensional case or with just one dimensional coordinate, such as the x coordinate. If this is the case, would the equation then become:
$ \frac{\partial c}{\partial t} = D\left(\frac{\partial^2 c}{\partial x^2}\right) + \left(\frac{\partial D}{\partial x} - v_1\right)\frac{\partial c}{\partial x} - c\left(\frac{\partial v_1}{\partial x}\right) + R $
Is this form correct? If so, how do I set it up for integration?