| Citation: |
Junsheng Duan, Randolph Rach, Dichen Hu. HIGH ACCURACY PIECEWISE-ANALYTIC SOLUTIONS AND HIGHER-ORDER NUMERIC SOLUTIONS OF PROJECTILE MOTION WITH A QUADRATIC DRAG FORCE BY THE MULTISTAGE MODIFIED DECOMPOSITION METHOD[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1679-1701. doi: 10.11948/20210277
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HIGH ACCURACY PIECEWISE-ANALYTIC SOLUTIONS AND HIGHER-ORDER NUMERIC SOLUTIONS OF PROJECTILE MOTION WITH A QUADRATIC DRAG FORCE BY THE MULTISTAGE MODIFIED DECOMPOSITION METHOD
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Abstract
We apply the multistage modified decomposition method (MDM) of Rach, Adomian, and Meyers to simulate the atmospheric projectile trajectory subject to a quadratic drag force. We readily obtain both the approximate analytic and numeric solutions through means of one-step recurrence algorithms based on the concept of analytic continuation, where the step size $ h $ and the order $ m $ are used to control errors. Simply put, the numeric solutions become the nodal values of our piecewise-analytic approximations. The realistic mathematical model includes sinusoidal, quadratic, reciprocal, and product nonlinearities which are conveniently treated by the Adomian polynomials without resort to any linearization or perturbation whatsoever. Fast algorithms of the Adomian polynomials guarantee the efficiency of our approach, and both the approximate analytic and numeric higher-order solutions can be readily generated at will unlike the usual Runge-Kutta methods that rely on a crude linearization. Multistage analytic and numeric decomposition algorithms demonstrate the rapid convergence of our new approach, where the MDM is based on the nonlinear transformation of series by the Adomian-Rach theorem. As an example, we also determine several important aerodynamic measures for the trajectory of a baseball such as the time of ascent, the velocity at the trajectory apex, the maximum height of ascent, then the flight range, the impact velocity and the impact angle with respect to the horizontal, the optimal launch angle, and the maximum flight range. We consider the error analyses for the multistage analytic approximations including the remainder error functions and also introduce the accumulative remainder error functions and the accumulative remainder error bounds for the numeric solutions. Our approximate solutions compare most favorably to the exact solution by Bernoulli.
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References
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Junsheng Duan, Randolph Rach, Dichen Hu. HIGH ACCURACY PIECEWISE-ANALYTIC SOLUTIONS AND HIGHER-ORDER NUMERIC SOLUTIONS OF PROJECTILE MOTION WITH A QUADRATIC DRAG FORCE BY THE MULTISTAGE MODIFIED DECOMPOSITION METHOD[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1679-1701. doi: 10.11948/20210277
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Figure 1.
Schematic diagram of a moving projectile.
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Figure 2.
Solution
$ \Phi _{v, m + 1}(t) $ $ 0\leq t\leq 12 $ -
Figure 3.
Remainder error function
$ REF_{v, m+1}(t) $ -
Figure 4.
Accumulative remainder error function
$ AREF_{v, m+1}(t) $ -
Figure 5.
Solution
$ \Phi _{\theta, m + 1}(t) $ $ 0\leq t\leq 12 $ -
Figure 6.
Remainder error function
$ REF_{\theta, m+1}(t) $ -
Figure 7.
Accumulative remainder error function
$ AREF_{\theta, m+1}(t) $ -
Figure 8.
Solution
$ \Phi _{x, m + 1}(t) $ $ 0\leq t\leq 12 $ -
Figure 9.
Remainder error function
$ REF_{x, m+1}(t) $ -
Figure 10.
Accumulative remainder error function
$ AREF_{x, m+1}(t) $ -
Figure 11.
Solution
$ \Phi _{y, m + 1}(t) $ $ 0\leq t\leq 12 $ -
Figure 12.
Remainder error function
$ REF_{y, m+1}(t) $ -
Figure 13.
Accumulative remainder error function
$ AREF_{y, m+1}(t) $ -
Figure 14.
Comparison of the exact value (solid line) and numeric results (dots) of
$ v $ $ \theta $ -
Figure 15.
Absolute errors of the numeric solutions
$ v^{\langle i\rangle} $ $ \theta $ -
Figure 16.
Absolute errors of the numeric solutions
$ v^{\langle i\rangle} $ $ t $ -
Figure 17.
Curve of the Cartesian coordinates
$ y $ $ x $ -
Figure 18.
Trajectories of a baseball, tennisball and shuttlecock for
$ v_0=40\; {\rm m/s} $ $ \theta_0=45^\circ $ -
Figure 19.
Schematic diagram for the flight range with the impact point on the horizontal line
$ y=y_1 $ -
Figure 20.
Plots of the horizontal flight range versus the launch angle
$ \theta_0 $ $ ^\circ $ $ ^\circ $ -
Figure 21.
Plots of the horizontal flight range versus the launch angle
$ \theta_0 $ $ ^\circ $ $ ^\circ $