[Paper Review] On some computational aspects of Hermite wavelets on a class of SBVPs arising in exothermic reactions

Amit K. Verma, Diksha Tiwari|arXiv (Cornell University)|Nov 3, 2019

Fractional Differential Equations Solutions27 references5 citations

TL;DR

This paper proposes a novel class of singular nonlinear boundary value problems (SBVPs) modeling exothermic reactions and introduces four computationally stable numerical methods using Hermite wavelets coupled with quasilinearization and Newton-Raphson iteration. The methods achieve high accuracy with few iterations and demonstrate numerical stability, outperforming Haar wavelet-based approaches in solving Lane-Emden-type equations and real-world SBVPs from thermal reaction dynamics.

ABSTRACT

We propose a new class of SBVPs which deals with exothermic reactions. We also propose four computationally stable methods to solve singular nonlinear BVPs by using Hermite wavelet collocation which are coupled with Newton's quasilinearization and Newton-Raphson method. We compare the results obtained with Hermite Wavelets with Haar wavelet collocation. The efficiency of these methods are verified by applying these four methods on Lane-Emden equations. Convergence analysis is also presented.

Motivation & Objective

• To model a new class of singular nonlinear boundary value problems (SBVPs) arising in exothermic reactions using a generalized form of the Arrhenius law.
• To address the computational challenge of solving nonlinear SBVPs with singularities and strong nonlinearity using wavelet-based numerical schemes.
• To develop and compare four novel numerical methods—Hermite Wavelet Newton Approach (HeWNA), Hermite Wavelet Quasilinearization Approach (HeWQA), Haar Wavelet Newton Approach (HWNA), and Haer Wavelet Quasilinearization Approach (HWQA)—for high-accuracy solutions.
• To establish convergence analysis for the HeWNA method and validate the methods on benchmark and real-life SBVPs.
• To demonstrate the numerical stability and robustness of the proposed methods under small perturbations in initial vectors.

Proposed method

• Employ Hermite wavelets via Multi-Resolution Analysis (MRA) to construct a basis for spectral collocation, enabling efficient approximation of solutions to nonlinear SBVPs.
• Apply quasilinearization to linearize the nonlinear terms in the differential equation, transforming the problem into a sequence of linear subproblems.
• Use Newton-Raphson iteration to solve the resulting nonlinear algebraic systems arising from wavelet collocation, ensuring fast convergence.
• Implement operational matrices of integration and differentiation for Hermite wavelets to convert differential equations into algebraic equations.
• Handle singularities at t=0 by leveraging the compact support and orthogonality of Hermite wavelets, which allow accurate resolution with minimal grid points.
• Compare results with Haar wavelet-based methods (HWNA and HWQA) to validate superior accuracy and stability of Hermite wavelet approaches.

Experimental results

Research questions

• RQ1 Can Hermite wavelets be effectively used to solve a new class of singular nonlinear SBVPs modeling exothermic reactions with improved accuracy compared to existing wavelet methods?
• RQ2 How do the proposed Hermite wavelet-based methods (HeWNA, HeWQA, HWNA, HWQA) compare in accuracy and convergence speed to established Haar wavelet methods?
• RQ3 To what extent does the proposed method maintain numerical stability under small perturbations in the initial guess vector?
• RQ4 Does the convergence analysis of the HeWNA method confirm that the error decreases as the resolution level J increases?
• RQ5 Can the proposed methods accurately solve real-world SBVPs such as the Lane-Emden equation and thermal distribution in the human head, where exact solutions are unknown?

Key findings

• The proposed Hermite wavelet-based methods (HeWNA and HeWQA) achieve higher accuracy than Haar wavelet-based counterparts (HWNA and HWQA), as shown in Table 11 and Figure 9 for Example 5.4.
• For Example 5.4 (nonlinear Lane-Emden-type SBVP), the computed solutions using HeWNA and HeWQA agree to at least 8 decimal places with the same results from HWNA and HWQA, indicating high consistency and stability.
• In Example 5.5 (thermal distribution in the human head), the HeWNA and HeWQA methods produce solutions that are more accurate than HWNA and HWQA, with differences in the 4th to 6th decimal places, as shown in Table 12.
• The convergence analysis confirms that the error norm ||E_{k,M}|| tends to zero as M → ∞, indicating that solution accuracy improves with increasing resolution level J.
• Small perturbations in the initial vector (e.g., [0.9,0.9,…,0.9] or [0.8,0.8,…,0.8]) do not significantly alter the solution, demonstrating numerical stability of the proposed methods.
• The methods successfully handle singularities and nonlinearities in SBVPs without requiring exact solutions, as validated on five real-life examples including those from Chambre, Nakamura, and Duggan-Goodman.