# New PDF release: A Treatise on Linear Differential Equations, Volume I:

By Thomas Craig

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Extra info for A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients

Example text

2 the normalized functions X i (x) and Yi (y) are illustrated for i = 1, . . , 4 and M = 101. One can observe that, as i increases, X i (x) and Yi (y) both account for a higher frequency content of the numerical solution. In Fig. 3, we show the 2D reconstructed PGD solution together with the normalized functions X i (x) and Yi (y). For clarity reasons the 2D solution is shown on a coarser grid. In the sequel we illustrate the convergence of the PGD solution towards the analytical solution as a function of both the discretization and the number of enrichment steps.

5 A Brief Overview of PGD Applications 19 identification or optimization can easily be handled. The PGD framework indeed allows one to perform these tasks very efficiently as simple post-processing operations acting on the PGD general solution computed once and for all. Inverse methods were addressed in [60] in the context of real-time simulations. They were coupled with control strategies in [61] as a first step towards DDDAS. Moreover, because the PGD parametric solution is pre-computed offline, it can be used online in real-time mode using light computing platforms like smartphones [49, 61].

Parallel time integration strategies, inspired from the ones proposed in [46], were considered in [74] within the PGD framework. The coupling of other multi-physics models in the context of composites manufacturing processes was performed in [56, 75]. Non-incremental parametric solutions defined in evolving domains were addressed in [76]. In [77], the solution of the Maxwell equations of electromagnetism was considered. Uncertainty quantification and propagation have been deeply considered in the works of A.