2,422 research outputs found
Wavelet Galerkin method for fractional elliptic differential equations
Under the guidance of the general theory developed for classical partial
differential equations (PDEs), we investigate the Riesz bases of wavelets in
the spaces where fractional PDEs usually work, and their applications in
numerically solving fractional elliptic differential equations (FEDEs). The
technique issues are solved and the detailed algorithm descriptions are
provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin
method we propose for FEDEs has the striking benefit of efficiency, since the
condition numbers of the corresponding stiffness matrixes are small and
uniformly bounded; and the Toeplitz structure of the matrix still can be used
to reduce cost. Numerical results and comparison with the ordinary Galerkin
methods are presented to demonstrate the advantages of the wavelet Galerkin
method we provide.Comment: 20 pages, 0 figure
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
We study linear parabolic initial-value problems in a space-time variational
formulation based on fractional calculus. This formulation uses "time
derivatives of order one half" on the bi-infinite time axis. We show that for
linear, parabolic initial-boundary value problems on , the
corresponding bilinear form admits an inf-sup condition with sparse tensor
product trial and test function spaces. We deduce optimality of compressive,
space-time Galerkin discretizations, where stability of Galerkin approximations
is implied by the well-posedness of the parabolic operator equation. The
variational setting adopted here admits more general Riesz bases than previous
work; in particular, no stability in negative order Sobolev spaces on the
spatial or temporal domains is required of the Riesz bases accommodated by the
present formulation. The trial and test spaces are based on Sobolev spaces of
equal order with respect to the temporal variable. Sparse tensor products
of multi-level decompositions of the spatial and temporal spaces in Galerkin
discretizations lead to large, non-symmetric linear systems of equations. We
prove that their condition numbers are uniformly bounded with respect to the
discretization level. In terms of the total number of degrees of freedom, the
convergence orders equal, up to logarithmic terms, those of best -term
approximations of solutions of the corresponding elliptic problems.Comment: 26 page
Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method
In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of θ and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.This research has been partially supported by Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-0971-B-100 and Fundación Séneca -Agencia de Ciencia y Tecnología de la Región de Murcia grant number 20783/PI/18. Also, It has been supported by the National Research Program for Universities (NRPU), Higher Education Commission, Pakistan, No. 8103/Punjab/NRPU/R and D/HEC/2017
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