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In this article, the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed. In an interesting study, Kloosterziel (Journal of Engineering Mathematics 24, 213-236 (1990)) represented a solution to an initial value problem (IVP) for the heat equation, with an initial data in a class of rapidly decaying functions, as a series of self-similar solutions to the heat equation. This approach quickly revealed the large time behaviour for the solution to the IVP. Inspired by Kloosterziel's approach, the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation. The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained.

In this article, we construct solutions of a nonhomogeneous Burgers equation subject to certain unbounded initial profiles. In an interesting study, Kloosterziel [1] represented the solution of an initial value problem (IVP) for the heat equation, with initial data in L 2 (R, e x 2 /2 ), as a series of the self-similar solutions of the heat equation. This approach quickly revealed the large time behavior for the solution of the IVP. Inspired by Kloosterziel [1]'s approach, we express the solution of the nonhomogeneous Burgers equation in terms of the self-similar solutions of a linear partial differential equation with variable coefficients. Finally, we also obtain the large time behavior of the solution of the nonhomogeneous Burgers equation.

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Accurate and consistent information on the area and production of field crops is vital for national and state planning and ensuring food security in India. Satellite-based remote sensing offers a suitable and cost-effective technique for regional- and national-scale crop monitoring. The use of remote sensing data for crop yield estimation has been demonstrated using a semi-physical approach with reasonable success. Assimilating remote sensing data with the DSSAT model and spectral indices-based regression analysis are promising methods for spatially estimating rice crop yields. Rice area and yield in the Cauvery delta zone of Tamil Nadu, India was estimated during samba (August–January) season in the years 2020–2021 using Sentinel 1A Synthetic Aperture Radar satellite data with three different spatial yield estimation methods, namely a spectral indices-based regression analysis, semi-physical approach, and integrating remote products with DSSAT crop growth model. A rice area map was generated for the study area using a rule-based classifier approach utilizing parameterization with a classification accuracy of 94.5% and a kappa score of 0.89. The total classified rice area in Cauvery Delta Region was 379,767 ha, and the Start of Season (SoS) maps for samba season revealed that the major planting period for rice was between 22 September and 9 November in 2020. The study also aimed to identify promising spatial yield estimation techniques for optimal rice yield prediction over large areas. Regression models resulted in rice yields of 3234 to 3905 kg ha−1 with a mean of 3654 kg ha−1. The net primary product was computed using the periodical PAR, fAPAR, Wstress, Tstress, and maximum radiation use efficiency in a semi-physical approach. The resultant rice yields ranged between 2652 and 3438 kg ha−1 with the mean of 3076 kg ha−1. During the integration of remote sensing products with crop growth models, LAI values were extracted from dB images and utilized to simulate rice yields in the range of 3684 to 4012 kg ha−1 with the mean of 3855 kg ha−1. When compared to the semi-physical approach, both integrating remote sensing products with the DSSAT crop growth model and spectral indices-based regression analysis had R2 greater than 0.80, NRMSE of less than 10%, and agreement of more than 90%, indicating that these two approaches could be used for spatial rice yield estimation.

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