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Theoretical Models for Investigating The Processes of Nanofilm Deposition onto Porous Templates of Aluminum Oxide
Published in Rishat G. Valeev, Alexander V. Vakhrushev, Aleksey Yu. Fedotov, Dmitrii I. Petukhov, A. N. Beltiukov, A. L. Trigub, A. V. Severyukhin, Nanostructured Semiconductors in Porous Alumina Matrices, 2019
Rishat G. Valeev, Alexander V. Vakhrushev, Aleksey Yu. Fedotov, Dmitrii I. Petukhov
The method (eq 6.72) is called the modified Euler method. The step error of modified Euler method is determined by the value O (Δt3), consequently, the method has second order of accuracy. The disadvantage of this method is the necessity to calculate partial derivatives of the function F¯(t,r¯,V¯)
Ordinary differential equations
Published in Edwin Zondervan, A Numerical Primer for the Chemical Engineer, 2019
The Euler method is a simple way of solving a differential equation. If the following ODE (ordinary differential equation) is given: dxdt=f(x,t) with the initial condition x(t = 0) = x0, we could generate an estimate of x at t + δt as x(t+δt)=x(t)+dxdtδt=x(t)+f(x,t)δt, so we can step forward in time, by evaluating the gradient, from the current step to the next step. Figure 7.1 shows how this looks graphically.
Computing Solutions of Ordinary Differential Equations
Published in Nayef Ghasem, Modeling and Simulation of Chemical Process Systems, 2018
The Euler method is a practical numerical method. The Euler method is a first-order numerical procedure for solving ODEs with a given initial value. Divide the region of interest [a,b] into discrete values of x=nh,n=0,1….N, spaced at intervals h=(b−a)/N. Use the forward difference approximation for the differential coefficient: () f(xn,yn)≈(yn+1−yn)h
Trajectory-following and off-tracking minimisation of long combination vehicles: a comparison between nonlinear and linear model predictive control
Published in Vehicle System Dynamics, 2023
Toheed Ghandriz, Bengt Jacobson, Peter Nilsson, Leo Laine
We first perform the discretisation and then the linearisation of the optimal control problem (4) for constructing an NMPC. Then, we solve the NMPC by using the SQP or LTV-MPC approaches [24]. We discretise the space to a equidistant grid with a step size assuming a zero-order-holder that maintains the controls constant within a discretisation step. The system derivatives must be approximated using a numerical method. While the explicit Euler method is the most straightforward approximation, it is inefficient and for a large step size it may become unstable. Multistep methods can also be used, but they require additional care for initialisation. On the other hand, Runge–Kutta methods are not suitable for implicit ODEs such as (1). Therefore, we used the Euler approximation in this paper, i.e. where, in this paper, for m, the numerical solution is stable, and the system Jacobian defined in later sections is not singular. Larger step size than 1 m might result in singular Jacobian and hence divergence of the numerical solution depending on the selected manoeuvrer.
Calculus of variations for estimation in ODE–PDE landslide-like models with discrete-time asynchronous measurements
Published in International Journal of Control, 2022
Mohit Mishra, Gildas Besançon, Guillaume Chambon, Laurent Baillet
The Euler method is based on a truncated Taylor series expansion (Ascher & Petzold, 1998), i.e. expansion of y in the neighbourhood of , where with , N being the number of time steps, gives At each time step, higher order terms are neglected which induces local truncation error proportional to the square of the step size , and the global error (error at a given time) is proportional to the step size. The value of variable y at time in Equation (1) is computed as The Euler method is used for numerical integration of system and adjoint ODEs, both forward and backward in time.
Adaptive neural control of vehicle yaw stability with active front steering using an improved random projection neural network
Published in Vehicle System Dynamics, 2021
Wei Huang, Pak Kin Wong, Ka In Wong, Chi Man Vong, Jing Zhao
After substituting Equations (3) to (6) into Equations (1) and (2), the vehicle-road system dynamics is then formulated in the discrete representation using Euler method with the sampling interval as follows: where is the system input which stands for the superposition angle ; is the output of the vehicle dynamic system (i.e. lateral vehicle speed and yaw rate ); and are weight parameters; is time step. Note that the stability of the Euler method can be guaranteed as long as the so-called Courant–Friedrichs-Lewy condition is fulfilled [31].