# Analysis of The Integration Algorithms for First Order, Parabolic Equations; Modal Decomposition - Lesson 7

The ODEs written using the Euler family will be analyzed here. Before performing the analysis, the ODE is written in a slightly different form. Basic properties of the integration algorithms are of interest, and it is useful to use the homogeneous ODEs which have forcing equal to zero. The stability and consistency of the time integration algorithms will be analyzed. Modal decomposition is performed which is invoking the related generalized Eigenvalue problem. The orthonormality property of the eigenvectors is stated.

The discussion continues from the previous video. Gram-Schmidt process is used here for the orthonormalization. Linearly independent vectors are chosen and orthonormalized. Modal coefficient of ‘d’ is written.