Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomogeneous case is solved without restrictions in boundary and initial conditions. The generalized Mittag-Leffler functions with three parameters are used and the solutions presented are expressed in terms of Wiman’s functions and their derivatives.
Using nonorthogonal bases in spectral methods demands considerable effort, because applying the Gram-Schmidt process is a fundamental condition for calculations. However, operational matrices numerical methods are being used in an increasing way and extensions for nonorthogonal bases appear, requiring simplified procedures. Here, extending previous work, an efficient tensorial method is presented, in order to simplify the calculations related to the use of nonorthogonal bases in spectral numerical problems. The method is calledcoadjoint formalismand is based onbracketDirac’s formulation of quantum mechanics. Some examples are presented, showing how simple it is to use the method.
This study proposes a diffeomorphic map for planar irregular domains, which transforms them into a unitary circle suitable for implementing spectral solutions on a complete and orthogonal basis. The domain must be enclosed by a Jordan curve that is univocally described in polar coordinates by a
C
2
class function. Because of the achieved precision, general eigenvalue problems can be solved, yielding hundreds of eigenvalues and eigenfunctions in a single shot with at least ten-digit precision. In a constructive approach, there is no need for special functions or roots. The examples focus on the Helmholtz equation, but the method applies to other eigenvalue problems such as Hamiltonians. For validation, we compare all the results with the values obtained by finite elements (FE), some with other studies, and discuss sharp error estimate criteria and the asymptotic behavior of the eigenvalues. To demonstrate the generality of the approach, we present several examples with different geometries in which solutions emerge naturally. Considering the spectral accuracy, half of the examples are unpublished solutions, and the other half agree with or expand the precision of previous studies using different approaches.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.