In this study we will consider moving mesh methods for solving one-dimensional time dependent PDEs. The solution and mesh are obtained simultaneously by solving a system of differential-algebraic equations. The differential equations involve the solution and the mesh, while the algebraic equations involve several geometrical variables such as $\theta$ (the tangent angle), $U$ (the normal velocity of the solution curve) and $T$ (tangent velocity). The equal-arclength principle is employed to give a close form for $T$. For viscous conservation laws, we prove rigorously that the proposed system of moving mesh equations is well-posed, in the sense that first order perturbations for the solution and mesh can be controlled by the initial perturbation. Several test problems are considered and numerical experiments for the moving mesh equations are performed. The numerical results suggest that the proposed system of moving mesh equations is appropriate for solving (stiff) time dependent PDEs.