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The multi-step backward difference formulas of order k (BDF-k) for 3 ≤ k ≤ 5 are proposed for solving the extended Fisher--Kolmogorov equation. Based upon the careful discrete gradient structures of the BDF-k formulas, the suggested numerical schemes are proved to preserve the energy dissipation laws at the discrete levels. The maximum norm priori estimate of the numerical solution is established by means of the energy stable property. With the help of discrete orthogonal convolution kernels techniques, the L2 norm error estimates of the implicit BDF-k schemes are established. Several numerical experiments are included to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

To overcome inadequate functional integration, irregular pits, and insufficient agricultural machinery and agronomy in tobacco seedling transplanting equipment under film, a digging transplanting integrated machine was built. The machine uses parallel four-bar mechanism. The excavation equipment uses conical spiral blades. Using the trajectory approach, the four bar mechanism's fundamental parameters can be reversed. This method meets the requirements for coordinated excavation and transplantation. Mechanical study of soil units determines upward movement characteristics. Additionally, spiral angle, spiral blade outer diameter, and spiral blade speed are quantified to affect soil movement. The range of factors is derived. Based on three factors binary regression orthogonal combination simulated optimization experiment, a discrete element analysis was performed to enhance parameters. Experimental factors optimized include a spiral angle of 19°, a spiral blade diameter of 378 mm, and a rotational speed of 258 rpm. The two optimized test indicators are: a 193 mm hole depth and 350 mm hole diameter. The improved results were verified by field validation studies using a 191 mm hole depth and 347 mm hole diameter. The simulation optimization results and field validation experiment results are very similar. This study provides a technical basis for tobacco farming concurrent drilling and transplanting research. [ABSTRACT FROM AUTHOR]