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The article is devoted to the application of multiple Fourier-Legendre seriesto implementation of strong exponential Milstein and Wagner-Platen methods fornon-commutative semilinear stochastic partial differential equations withmultiplicative trace class noise. These methods have strong orders ofconvergence $1.0-\varepsilon$ and $1.5-\varepsilon$ correspondingly (here$\varepsilon$ is an arbitrary small positive real number) with respect to thetemporal discretization. The theorem on mean-square convergence ofapproximations of iterated stochastic integrals of multiplicities 1 to 3 withrespect to the infinite-dimensional $Q$-Wiener process is formulated andproved. The results of the article can be applied to implementation ofexponential Milstein and Wagner-Platen methods for non-commutative semilinearstochastic partial differential equations with multiplicative trace classnoise.

Application of Multiple Fourier-Legendre Series to Implementation of  Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative  Semilinear Stochastic Partial Differential Equations