This paper addresses an efficient approach based on Haar wavelets to solve multi-dimensional nonlinear partial differential equations (PDEs). The proposed Haar wavelets method (PHWM) approximates the highest-order partial derivatives in the governing equation by utilizing the Haar wavelet series. These series are then integrated within the given limits of integration to derive lower-order partial derivatives and the Haar wavelet solutions. Then, we express the derivatives of the wavelet solution u in terms of u itself by eliminating the unknown coefficients through the wavelet solution expressions. This technique can be effectively applied to scenarios involving nonlinear problems and is more efficient and easier to implement compared to conventional Haar wavelet methods. Unlike the conventional Haar wavelets methods, the PHWM requires the inversion of a coefficient matrix of size <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(2^{J+1})^d \times (2^{J+1})^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow> <mi>J</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </msup> <mo>×</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow> <mi>J</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where d is the dimensional parameter and J is the maximum level of resolution of the Haar wavelet. The PHWM has the same order of convergence as the conventional Haar wavelet method; that is, the PHWM also has a second order of convergence. We demonstrate the effectiveness of our approach through multiple examples in one-, two-, and three-dimensional spaces. In the current work, we also use the new splitting technique in the wavelet method, introduced recently in Liu et al. (Eng. Anal. Bound. Elem. 125, 124–134, 2021), to linearize the nonlinear PDEs. Our experimental results validate the accuracy and efficiency of the proposed algorithms, rendering them highly suitable for solving high-dimensional elliptic PDEs among wavelet methods.

An efficient numerical algorithm based on Haar wavelets for multi-dimensional nonlinear elliptic PDEs

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发表日期：

2025年3月25日

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Masood Ahmad,
Muhammad Ahsan,
Zaheer Uddin

摘要

This paper addresses an efficient approach based on Haar wavelets to solve multi-dimensional nonlinear partial differential equations (PDEs). The proposed Haar wavelets method (PHWM) approximates the highest-order partial derivatives in the governing equation by utilizing the Haar wavelet series. These series are then integrated within the given limits of integration to derive lower-order partial derivatives and the Haar wavelet solutions. Then, we express the derivatives of the wavelet solution u in terms of u itself by eliminating the unknown coefficients through the wavelet solution expressions. This technique can be effectively applied to scenarios involving nonlinear problems and is more efficient and easier to implement compared to conventional Haar wavelet methods. Unlike the conventional Haar wavelets methods, the PHWM requires the inversion of a coefficient matrix of size $(2^{J+1})^d \times (2^{J+1})^d$ ${(2^{J + 1})}^{d} \times {(2^{J + 1})}^{d}$ , where d is the dimensional parameter and J is the maximum level of resolution of the Haar wavelet. The PHWM has the same order of convergence as the conventional Haar wavelet method; that is, the PHWM also has a second order of convergence. We demonstrate the effectiveness of our approach through multiple examples in one-, two-, and three-dimensional spaces. In the current work, we also use the new splitting technique in the wavelet method, introduced recently in Liu et al. (Eng. Anal. Bound. Elem. 125, 124–134, 2021), to linearize the nonlinear PDEs. Our experimental results validate the accuracy and efficiency of the proposed algorithms, rendering them highly suitable for solving high-dimensional elliptic PDEs among wavelet methods.