Comparative Analysis of He's and Daftardar-Jafari Polynomials for Solving Non-Linear Fractional Partial Differential Equations Using the Iterative Laplace Transform Method

Core Concepts

He's and Daftardar-Jafari polynomials, combined with the Iterative Laplace Transform Method (ILTM), offer a highly accurate and efficient approach to solving non-linear fractional partial differential equations, outperforming existing techniques in accuracy.

Abstract

• Bibliographic Information: Khan, Q., & Suen, A. (2024). Comparative Analysis of Polynomials with Their Computational Costs. arXiv preprint arXiv:2411.00487v1.
• Research Objective: This paper aims to compare the effectiveness of He's and Daftardar-Jafari polynomials in solving non-linear fractional partial differential equations (FPDEs) using the Iterative Laplace Transform Method (ILTM).
• Methodology: The researchers apply ILTM with both He's and Daftardar-Jafari polynomials to three distinct non-linear FPDEs previously addressed in the literature. They compare the accuracy of their solutions against existing results using absolute error analysis at various iterations, spatial points, and time levels.
• Key Findings: The study demonstrates that both polynomial methods, when coupled with ILTM, provide accurate solutions for the chosen FPDEs. Notably, Daftardar-Jafari polynomials exhibit superior accuracy compared to He's polynomials in most test cases. The authors provide graphical and tabular representations of their findings, highlighting the convergence and accuracy advantages of their approach.
• Main Conclusions: The combination of ILTM with either He's or Daftardar-Jafari polynomials offers a robust and accurate method for solving non-linear FPDEs. The authors suggest that this approach can be extended to a wider range of fractional problems, opening avenues for future research.
• Significance: This research contributes to the field of scientific computing by providing a more accurate and efficient method for solving complex mathematical models represented by non-linear FPDEs. These equations have broad applications in various scientific and engineering disciplines.
• Limitations and Future Research: The study focuses on a limited set of FPDEs. Further research could explore the effectiveness of this method on a wider range of fractional problems, including fractional diffusion equations, fractional wave equations, and fractional Schrödinger equations. Additionally, investigating the computational cost and efficiency of these polynomial methods in comparison to other existing techniques would be beneficial.

Stats

Absolute error comparisons are provided for different iterations (k = 1 to 5 or 6) at specific spatial points (x = 0.1, 0.3, 0.5, 0.7, 0.9) and time levels (t = 0.1, 0.3, 0.5, 0.7, 0.9 for problems 1 and 2; t = 0.01, 0.03, 0.05, 0.07, 0.09 for problem 3).
Problem 3 includes a comparison of solution accuracy at t = 0.001 for He's polynomials, Daftardar-Jafari polynomials, NIM, and OAFM methods against the exact solution.

Quotes

"In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations."
"Comparative analysis with existing techniques reveals that our approach yields more precise solutions."
"The results, presented through graphs and tables, indicate that He’s and Daftardar-Jafari polynomials significantly enhance accuracy."
"Due to its straightforward implementation, [the] proposed method can be extended for application to a broader range of problems."

Key Insights Distilled From

Comparative Analysis of Polynomials with Their Computational Costs

by Qasim Khan, ... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00487.pdf

Deeper Inquiries

How does the computational cost of ILTM with He's and Daftardar-Jafari polynomials compare to other numerical methods for solving non-linear FPDEs, such as finite difference methods or finite element methods?

Directly comparing the computational cost of ILTM using He's and Daftardar-Jafari polynomials with traditional numerical methods like finite difference methods (FDM) or finite element methods (FEM) for non-linear FPDEs is not straightforward. This is because they differ significantly in their underlying approaches and the types of problems they are best suited for.
Here's a breakdown:

FDM and FEM: These methods are based on discretizing the spatial and temporal domains, transforming the continuous FPDE into a system of algebraic equations. The computational cost heavily depends on the discretization scheme, mesh size, and the dimensionality of the problem. Finer meshes generally lead to higher accuracy but at the cost of increased computational time and memory usage.

ILTM with Polynomial Approximations: This approach leverages the Laplace transform to convert the FPDE into an algebraic equation in the Laplace domain. The non-linear terms are then handled using polynomial expansions (He's or Daftardar-Jafari). The computational cost is primarily associated with:

Laplace Transforms and Inverse Transforms: The efficiency of these operations depends on the complexity of the transformed equations and the algorithms used.
Polynomial Expansions: The number of terms included in the polynomial expansion directly impacts the accuracy and computational cost. Higher-order expansions improve accuracy but require more computations.
General Observations:

Problem Suitability: FDM and FEM are generally well-suited for problems with complex geometries and boundary conditions, where they provide flexibility in meshing. ILTM, on the other hand, might be advantageous for problems where analytical solutions in the Laplace domain are attainable or for specific types of non-linearities that lend themselves well to polynomial approximations.

Convergence Rate: ILTM, when it converges, often exhibits a faster convergence rate compared to FDM or FEM, potentially requiring fewer iterations to achieve a desired accuracy. However, the convergence of ILTM is not always guaranteed and depends on the nature of the non-linearity and the chosen polynomial.

Programming Complexity: Implementing ILTM with polynomial approximations might involve more analytical derivations and symbolic computations compared to the relatively standardized implementations of FDM and FEM.
In Conclusion:
There is no universally superior method. The choice between ILTM and traditional numerical methods depends on the specific non-linear FPDE, desired accuracy, computational resources, and the trade-off between programming effort and computational efficiency.

Could there be specific types of non-linear FPDEs where He's polynomials might outperform Daftardar-Jafari polynomials, or are the latter consistently superior in terms of accuracy?

While the provided context suggests that Daftardar-Jafari (D-J) polynomials might generally exhibit better accuracy compared to He's polynomials for the tested problems, it's premature to conclude that D-J polynomials are consistently superior. The relative performance of these polynomial approximation methods in the context of ILTM can be problem-specific, depending on the nature of the non-linearity.
Here's a nuanced perspective:

He's Polynomials: These polynomials are derived from a homotopy perturbation method and are known to be computationally efficient, especially for weakly non-linear problems. They might outperform D-J polynomials in cases where the non-linear term has a simpler structure, allowing for faster convergence with fewer terms in the expansion.

Daftardar-Jafari Polynomials: These polynomials are defined recursively and are designed to handle a broader class of non-linear operators. They might be more suitable for strongly non-linear FPDEs or cases where the non-linear term involves fractional derivatives or integrals, as they can capture more complex behavior.
Potential Scenarios for He's Polynomials to Outperform:

Weakly Non-linear FPDEs: When the non-linear term has a relatively small contribution to the overall solution behavior, He's polynomials, with their computational efficiency, might converge faster.

Specific Non-linear Forms: There might be specific functional forms of the non-linear term where the structure of He's polynomials aligns better, leading to faster convergence.
Factors Influencing Performance:

Choice of Initial Guess: The initial guess in the ILTM procedure can influence the convergence rate for both polynomial methods. A good initial guess can significantly speed up convergence.

Fractional Order (α): The fractional order of the derivatives in the FPDE can affect the convergence characteristics of both methods.
In Conclusion:
A definitive statement about the consistent superiority of one polynomial method over the other is not possible without extensive experimentation across a diverse range of non-linear FPDEs. The choice should be guided by the specific problem, the nature of the non-linearity, and potentially, preliminary numerical tests comparing the convergence behavior of both methods.

Considering the increasing use of fractional calculus in modeling complex systems, what are the potential implications of having more accurate and efficient numerical solvers for FPDEs in fields like materials science, finance, or climate modeling?

The development of more accurate and efficient numerical solvers for fractional partial differential equations (FPDEs) holds significant implications for various fields grappling with complex systems, including materials science, finance, and climate modeling. Here's an exploration of the potential impact:
1. Materials Science:

Enhanced Material Design: FPDEs can model anomalous diffusion processes, which are prevalent in materials with complex microstructures. Accurate solvers could lead to the design of materials with tailored properties, such as enhanced drug delivery systems or more efficient solar cells.

Predictive Modeling of Material Behavior: Understanding material fatigue, crack propagation, and other complex phenomena often involves FPDEs. Improved solvers would enable more reliable predictions of material behavior under various conditions, leading to safer and more durable structures.
2. Finance:

Accurate Option Pricing: Fractional models have gained traction in option pricing by incorporating long-range dependencies and memory effects in financial markets. Efficient solvers would facilitate more realistic and accurate option pricing models, leading to better risk management and investment strategies.

Improved Risk Assessment: FPDEs can model extreme events and jumps in financial markets, which are crucial for risk assessment. Accurate solvers would provide financial institutions with better tools to quantify and manage risk, contributing to financial stability.
3. Climate Modeling:

Realistic Climate Projections: FPDEs can capture anomalous diffusion of heat and pollutants in the atmosphere and oceans, providing a more accurate representation of climate dynamics. Efficient solvers would enhance the reliability of climate models, leading to more precise climate projections and better-informed policy decisions.

Modeling of Extreme Events: FPDEs can model extreme weather events, such as hurricanes and droughts, which are becoming more frequent due to climate change. Accurate solvers would improve our understanding and prediction of these events, enabling better preparedness and mitigation strategies.
Overall Implications:

Deeper Insights into Complex Systems: More accurate solvers would provide researchers with a powerful tool to gain deeper insights into the behavior of complex systems, leading to breakthroughs in various fields.

Improved Decision-Making: Accurate and efficient FPDE solvers would enable more informed decision-making in areas such as material design, financial investments, and climate change mitigation.

Technological Advancements: The development of these solvers would drive innovation in computational mathematics and computer science, leading to the creation of more powerful algorithms and software tools.
In Conclusion:
The advancement of numerical solvers for FPDEs has the potential to revolutionize our understanding and ability to model complex systems. This progress would have far-reaching implications, leading to technological advancements, improved decision-making, and a deeper understanding of the world around us.