A particular class of planar two-patch geometries, called bilinear-like G2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and ...possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations.
We investigate the space of C2-smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C2-smoothness of an isogeometric function and the G2-smoothness of its graph surface (cf. Groisser and Peters (2015) and Kapl et al. (2015). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G2 two-patch parameterizations. Numerical results obtained by performing L2-approximation and solving Poisson’s equation indicate that already the subspace possesses optimal approximation properties.
•Introduction of bilinear-like G2 two-patch parameterizations .•Investigation of the C2 smooth isogeometric space over this specific geometry.•Computation of dimension of the space.•Construction of a simple explicitly given and well-conditioned basis of the space.•Numerical experiments show optimal rates of convergence.
In this article, a new dynamical system equation is constructed, named the (3+1)-dimensional Hirota-bilinear-like equation. The new ‘like’ equation has more nonlinear terms than the original equation ...while they have the same bilinear form. The generalized Hirota bilinear operator and its logarithmic transformation are applied to obtain the new ‘like’ equation and to find its solutions. Some different types of symbolic solutions are obtained, which are the lump solutions and the lump-kink-type rogue wave solutions. The propagation images of the lump waves and the interaction which is between the lump-type waves and two-kink soliton-waves are simulated to analysis the shapes and movements of these solutions.
•A new (3+1)-dimensional Hirota-bilinear-like equation is constructed.•The bilinear form was constructed based on the generalized Hirota bilinear operator.•The lump solutions and the lump-kink-type solutions are obtained and plotted.
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