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Mesh-free Mixed Finite Element Approximation for Nonlinear Time-Fractional Biharmonic Problem Using Weighted B-Splines

Core Concepts
Proposal of a mesh-free method for solving a nonlinear time-fractional biharmonic problem using weighted b-splines.
The article introduces a fully-discrete scheme for solving a nonlinear time-fractional biharmonic problem using weighted b-splines. It converts the problem into an equivalent system, discretizes spatially and temporally, and combines computational benefits of b-splines and mesh-based elements. The proposed method provides smooth approximations with few parameters, enforces essential boundary conditions accurately, and exhibits superior convergence rates compared to previous schemes. The paper includes theoretical findings supported by numerical experiments validating the proposed method's advantages.
Dαt u + ∆2u - ∆u = f(u) in Σ, Dαt u(x,t) ∶= 1/Γ(1 - α) ∫0(t - s)^-α ∂u(x,s)/∂s ds for 0 < α < 1. Authors considered various PDEs including linear time-fractional biharmonic problems. Proposed scheme based on L2-1σ approximation and weighted b-splines. Error estimates derived for the fully-discrete scheme.

Deeper Inquiries

How does the proposed mesh-free method compare to traditional finite element methods

The proposed mesh-free method using weighted b-splines offers several advantages over traditional finite element methods. Firstly, it eliminates the need for a structured mesh, making it more flexible and adaptable to complex geometries. This is particularly beneficial in problems where generating a mesh is challenging or time-consuming. Additionally, the use of weighted b-splines allows for better approximation accuracy and smoother solutions compared to standard basis functions used in traditional finite element methods. The combination of these factors results in a more efficient and accurate numerical solution.

What are the practical implications of using weighted b-splines in solving nonlinear PDEs

Using weighted b-splines in solving nonlinear partial differential equations (PDEs) has several practical implications. One key advantage is that weighted b-splines can conform to essential boundary conditions accurately, which is crucial for obtaining reliable solutions to PDEs with complex boundary constraints. Furthermore, the stability issues often associated with standard b-spline basis functions are addressed through the use of weight functions and extension procedures in weighted b-splines. Moreover, by employing weighted b-splines, one can achieve high-order accuracy in spatial approximations with fewer degrees of freedom compared to traditional methods. This leads to computational efficiency without compromising on solution quality. Overall, the practical implications include improved accuracy, stability, and flexibility when dealing with nonlinear PDEs.

How can the findings of this study be applied to other types of fractional differential equations

The findings from this study on solving nonlinear time-fractional biharmonic problems using weighted b-splines can be extended and applied to various other types of fractional differential equations (FDEs). The methodology developed here showcases how mesh-free mixed finite element approximation techniques can effectively handle FDEs with Caputo derivatives. These techniques could be adapted and implemented for different classes of fractional differential equations such as fractional diffusion equations, fractional wave equations, or even multidimensional fractional models. By adjusting parameters like degree of splines or choice of weight functions based on specific characteristics of different FDEs, similar robust numerical schemes could be developed for diverse applications across various scientific fields involving fractional calculus.