Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique

Generalized polynomial chaos (gPC) is a spectral technique in random space to represent random variables and stochastic processes in terms of orthogonal polynomials of the Askey scheme. One of its most fruitful applications consists of solving random differential equations. With gPC, stochastic solutions are expressed as orthogonal polynomials of the input random parameters. Different types of orthogonal polynomials can be chosen to achieve better convergence. This choice is dictated by the key correspondence between the weight function associated to orthogonal polynomials in the Askey scheme and the probability density functions of standard random variables. Otherwise, adaptive gPC constitutes a complementary spectral method to deal with arbitrary random variables in random differential equations. In its original formulation, adaptive gPC requires that both the unknowns and input random parameters enter polynomially in random differential equations. Regarding the inputs, if they appear as non-polynomial mappings of themselves, polynomial approximations are required and, as a consequence, loss of accuracy will be carried out in computations. In this paper an extended version of adaptive gPC is developed to circumvent these limitations of adaptive gPC by taking advantage of the random variable transformation method. A number of illustrative examples show the superiority of the extended adaptive gPC for solving nonlinear random differential equations. In addition, for the sake of completeness, in all examples randomness is tackled by nonlinear expressions.