Efficient Numerical Approximation of Nonlinear Fractional Order Boundary Value Problems using Weighted Residual Methods

Core Concepts

The authors introduce the application of weighted residual methods, namely Galerkin, Least Square, and Collocation, for efficiently finding approximate solutions to nonlinear fractional order boundary value problems.

Abstract

The content starts by providing an overview of fractional derivatives and weighted residual methods. It then formulates the application of Galerkin, Least Square, and Collocation methods for solving nonlinear fractional order boundary value problems.
The key highlights are:

Fractional derivatives in the Caputo sense are used to model the nonlinear fractional order differential equations.
Modified Legendre and Bernoulli polynomials are employed as weight functions in the weighted residual methods.
The Galerkin, Least Square, and Collocation methods are elaborately described for the numerical solution of the nonlinear fractional order boundary value problems.
Three test problems are considered to demonstrate the accuracy and reliability of the proposed methods by computing the absolute errors and comparing with the exact solutions.
The results show that the proposed methods provide efficient and accurate approximations to the nonlinear fractional order boundary value problems.

Stats

The authors provide the following key figures to support their analysis:

The absolute errors obtained using modified Legendre and Bernoulli polynomials of degree n=3 for the first two test problems.
The L∞ and L2 errors for the three test problems.

Quotes

"The method of weighted residuals seeks an approximate solution of the form ỹ(x) = ∑ ai Pi(x) / n i=1 where ỹ(x) denote the approximate solution can be expressed as the product of ai unknown, constant parameters to be determined and Pi(x) are weight functions."
"The Galerkin weighted residual (GWR) method using modified Legendre and Bernoulli polynomials as basis functions, we shall denote approximate trial solution by ũ(x) by considering u(x) denotes the exact solution to a boundary value problem."

Key Insights Distilled From

Approximation of Some Nonlinear Fractional Order BVPs by Weighted Residual Methods

by Umme Ruman,M... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03338.pdf

Deeper Inquiries

How can the proposed weighted residual methods be extended to solve higher-order nonlinear fractional order boundary value problems

The proposed weighted residual methods can be extended to solve higher-order nonlinear fractional order boundary value problems by increasing the complexity of the trial functions and weight functions. For higher-order problems, the trial functions can be expanded to include more terms, allowing for a more accurate representation of the solution. Additionally, higher-order weight functions can be utilized to capture the intricacies of the problem more effectively. By incorporating these adjustments, the methods can handle the increased complexity and nonlinearity of higher-order problems while maintaining accuracy and efficiency.

What are the limitations of the Galerkin, Least Square, and Collocation methods, and how can they be addressed to further improve the accuracy and efficiency

The limitations of the Galerkin, Least Square, and Collocation methods primarily revolve around the choice of basis functions and the computational complexity involved in solving the resulting systems of equations. To address these limitations and improve accuracy and efficiency, several strategies can be implemented.

Adaptive Basis Functions: Utilizing adaptive basis functions that can adjust to the problem's characteristics can enhance the accuracy of the solutions.
Error Estimation: Implementing error estimation techniques to refine the solution iteratively can improve accuracy without significantly increasing computational cost.
Parallel Computing: Leveraging parallel computing techniques can reduce the computational burden and speed up the solution process.
Optimization Algorithms: Incorporating optimization algorithms to optimize the choice of basis functions and parameters can lead to more accurate solutions.
By addressing these limitations and implementing these strategies, the Galerkin, Least Square, and Collocation methods can be enhanced to provide more accurate and efficient solutions for nonlinear fractional order boundary value problems.

Can the weighted residual methods be combined with other numerical techniques, such as finite element or spectral methods, to solve more complex nonlinear fractional order boundary value problems

The weighted residual methods can indeed be combined with other numerical techniques, such as finite element or spectral methods, to solve more complex nonlinear fractional order boundary value problems. By integrating these methods, the strengths of each approach can be leveraged to tackle the challenges posed by complex problems.

Finite Element Method: By combining the weighted residual methods with the finite element method, the solution domain can be discretized into smaller elements, allowing for more localized approximations and better handling of complex geometries.
Spectral Methods: Spectral methods can provide highly accurate solutions by representing the solution as a sum of basis functions with unknown coefficients. Combining spectral methods with weighted residual methods can enhance the accuracy and efficiency of the solution.
Hybrid Approaches: Hybrid approaches that integrate different numerical techniques can offer a comprehensive solution strategy for complex problems, leveraging the strengths of each method to overcome specific challenges.
By combining weighted residual methods with other numerical techniques, researchers and practitioners can tackle more intricate and challenging nonlinear fractional order boundary value problems effectively.