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A localized approach for nonlinear high-dimensional generalized Benjamin-Bona-Mahony-Burgers equation on regular and irregular domains.

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Title: A localized approach for nonlinear high-dimensional generalized Benjamin-Bona-Mahony-Burgers equation on regular and irregular domains.
Authors: Balootaki, Parisa Ahmadi1 (AUTHOR) p.ahmadi.b@khuisf.ac.ir, Fardi, Mojtaba2 (AUTHOR) m.fardi@sku.ac.ir, Azarnavid, Babak3 (AUTHOR) babakazarnavid@ubonab.ac.ir
Source: Soft Computing - A Fusion of Foundations, Methodologies & Applications. Apr2026, Vol. 30 Issue 4, p2175-2188. 14p.
Subjects: Partition of unity method, Radial basis functions, Partial differential equations, Numerical analysis
Abstract: The partition of unity method (PUM) based on radial basis functions (RBFs) is a highly effective localized approach that has recently been used to solve high-dimensional initial–boundary-value problems. This method computes local approximations within subdomains and then integrates them using unit functions to obtain a global approximation. Its efficacy relies on partitioning the primary domain into overlapping subdomains, from which a global approximation is derived through the linear combination of these local approximations. This paper investigates a direct PUM (DPUM) based on RBFs to address the nonlinear high-dimensional generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation on both convex and nonconvex domains. We employ polyharmonic splines to achieve local approximations. This direct approach eliminates the need to compute derivatives of weight functions, thereby enhancing computational efficiency compared to the standard PUM (SPUM). Consequently, the implementation of this method significantly reduces computational costs. Several numerical tests are performed to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:The partition of unity method (PUM) based on radial basis functions (RBFs) is a highly effective localized approach that has recently been used to solve high-dimensional initial–boundary-value problems. This method computes local approximations within subdomains and then integrates them using unit functions to obtain a global approximation. Its efficacy relies on partitioning the primary domain into overlapping subdomains, from which a global approximation is derived through the linear combination of these local approximations. This paper investigates a direct PUM (DPUM) based on RBFs to address the nonlinear high-dimensional generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation on both convex and nonconvex domains. We employ polyharmonic splines to achieve local approximations. This direct approach eliminates the need to compute derivatives of weight functions, thereby enhancing computational efficiency compared to the standard PUM (SPUM). Consequently, the implementation of this method significantly reduces computational costs. Several numerical tests are performed to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]
ISSN:14327643
DOI:10.1007/s00500-025-11032-w