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Comparative Analysis of Polynomials with Their Computational Costs

Qasim Khan, Anthony Suen

TL;DR

The results indicate that He's and Daftardar-Jafari polynomials significantly enhance accuracy in solving non-linear time and space fractional partial differential equations.

Abstract

In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations. We first outline the general methodology and then apply it to five distinct experiments. The proposed method, noted for its simplicity, demonstrates a high degree of accuracy. 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. Additionally, we provide an in-depth discussion on the computational costs associated with these polynomials. Due to its straightforward implementation, the proposed method can be extended for application to a broader range of problems.

Comparative Analysis of Polynomials with Their Computational Costs

TL;DR

The results indicate that He's and Daftardar-Jafari polynomials significantly enhance accuracy in solving non-linear time and space fractional partial differential equations.

Abstract

In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations. We first outline the general methodology and then apply it to five distinct experiments. The proposed method, noted for its simplicity, demonstrates a high degree of accuracy. 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. Additionally, we provide an in-depth discussion on the computational costs associated with these polynomials. Due to its straightforward implementation, the proposed method can be extended for application to a broader range of problems.

Paper Structure

This paper contains 9 sections, 1 theorem, 37 equations, 8 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries Concept and Methodology
  3. Basic Definitions
  4. Research Methodology
  5. Experimental Results
  6. Results, Discussion and Computational Cost
  7. Results and Discussion
  8. Computational Cost
  9. Conclusion

Key Result

Lemma 1

Let $f({{{t}}})$ and $g({{{t}}})$ be two piecewise continuous functions defined on the interval $[0, \infty)$, and these functions are of exponential order. Let $F(s) = {\mathcal{L}}[f({{{t}}})]$ and $G(s) = {\mathcal{L}}[g({{{t}}})]$, where ${\mathcal{L}}$ denotes the Laplace transform. Additionall

Figures (8)

  • Figure 1: Comparison of He's and D-J polynomials across different iterations, along with their associated absolute errors for problem \ref{['problem1']}.
  • Figure 2: Comparison of He's and D-J polynomials across different iterations, along with their associated absolute errors for problem \ref{['problem2']}.
  • Figure 3: Comparison of He's and D-J polynomials across different iterations, along with their associated absolute errors for problem \ref{['problem3']}.
  • Figure 4: Comparison of He's and D-J polynomials across different iterations, along with their associated absolute errors for problem \ref{['problem4']}.
  • Figure 5: Comparison of He's and D-J polynomials across different iterations, along with their associated absolute errors for problem \ref{['problem5']}.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1