The nonlinear calculus model can more accurately describe and characterize many physical phenomena in the engineering field than the traditional calculus model. The research and application fields of nonlinear calculus are deepening and expanding. The study of accurate and reliable solutions for nonlinear differential equations has always been favored by researchers . The existing methods for solving nonlinear differential equations mainly include analytical and numerical approximation solutions. But they each have their pros and cons. The circuit simulation solution method uses the fractional reactance approximation circuit to construct the calculus operator
The Bagley-Torvik equation is a typical nonlinear differential equation with two derivative terms. This method aims to demonstrate the shear stress-induced at any point in the fluid . The model can be directly represented by the nonlinear time derivative of the fluid velocity distribution.
The initial value problem of the Bagley-Torvik equation with inhomogeneous constant coefficients is:
In formula (1),
The classical solutions of nonlinear differential equations include Grünwald - Letnikov numerical approximate solution method and Green's function solution method (analytical solution).
We take the fractional
The second-order approximation Grünwald coefficient
And obtained by
In formula (9), * is the convolution operator. We use the generalized binomial method to calculate the first-order approximation Grünwald coefficients. Calculated as follows.
The first-order approximation numerical solution uses Grünwald - Letnikov equations (5) and (6) to approximate the second-order derivatives
From this, the first-order approximation numerical solution algorithm is derived as follows.
Thus, the first-order approximation numerical solution
In Eq. (13),
From this, we deduce the second-order approximation numerical solution algorithm
Thus we obtain the second-order approximation numerical solution
For equation (1), the reference describes and proves Green's function solution method in detail. Using the Green's function solution method, the analytical solution of equation (1) is obtained as
Where Green's function
Do a double infinite series define a special function. In the specific numerical calculation, only the truncated approximate form can be taken as follows
Then the corresponding approximate analytical solution
According to the algorithm formulas (12), (15), and (20) and using Matlab programming to solve the first-order approximation numerical solution, the second-order approximation numerical solution, and the approximate analytical solution. The result is shown in Figure 1.
By comparing these two classical nonlinear differential equation solving methods, it can be concluded that: (1) Green's function method is theoretically convergent. But the result of its operation will be related to the computer system . This results in a non-convergence condition for a tail
The circuit simulation method uses the fractional reactance approximation circuit to construct the nonlinear calculus operator
The formula for the Laplace transform of the Grünwald - Letnikov fractional derivative is as follows.
We take the one-sided Laplace transform of equation (1) to get the differential operator
So equation (22) can be transformed into
Its corresponding time-domain expression is
It can be seen from Figure 2 that the circuit simulation system is a negative feedback system. The feedback system has the characteristics of high stability and small error. This ensures the stability and reliability of solving the Bagley-Torvik equation by the circuit simulation method . The actual physical circuit system built according to this schematic diagram can solve nonlinear differential equations in real-time.
The circuit element that realizes the operation function of the calculus operator
In the formula,
The Carlson scale fractal lattice anti-approximation circuit is shown in Figure 3.
According to the iterative circuit in the figure, the input impedance function
In the formula,
Matlab Simulink platform has powerful functions. SimulinkLibrary Browser Browser contains common toolboxes, such as DSPSystem Toolbox, Control System Toolbox, Communications System Toolbox, etc. Various arithmetic modules are provided in each toolbox. We can use these encapsulated modules to build simulation block diagrams to solve differential equations . The parameters of these modules need to be set before calculating the solution.
First, set the parameters of the Simulink simulation platform. Set the simulation start time to 0s in the Simulations-Mode Configuration Parameter Parameter. The end time is the 30s. The maximum step size is 0.1. Solver selects ode45. ode45 indicates that the fourth-order to fifth-order Runge-Kutta algorithm is used. It uses a 4th-order method to provide candidate solutions and a 5th-order method to control the error . It is a numerical solution method of ordinary differential equations with adaptive step size.
Next, we set the input and output. The Step function block connected to the input generates a step signal. We set its initial step time to 1s. The initial value of the simulation is 8. The simulation end value is 0. In this way, a function similar to Eq. (2) can be generated. We let the image displayed by the oscilloscope follow. The image drawn by the m file is drawn in the same figure. We need to set the parameters of the oscilloscope. The article will set the history of the parameter column of the oscilloscope . At the same time we save the data to the workspace, set the variable name, and save format to Array. In this way, the output result y(t) can be called in the m file, and its image can be drawn on the figure. The transfer function module TransferFcn is used to characterize the transfer function. The specific transfer function expression is:
Set up the circuit parameters of the Carlson scale fractal lattice fractional reactance circuit according to the simulation environment built in Figure 4:
Figure 5 compares the solution results of three numerical solutions to the Bagley-Torvik equation . The circuit simulation method can accurately solve nonlinear differential equations with constant coefficients.
There is a calculus order
From the above results, we can draw the following conclusions: 1) The circuit simulation method can solve any nonlinear differential equation. 2) The frequency of the input function has a great influence on the solution result of the circuit simulation method. 3) In Different operation orders
In this paper, the Grünwald numerical approximation method and the Green's function method are introduced first. Then we elaborate the circuit simulation method and solve the Bagley-Torvik equation. Compared with the traditional numerical method, the solution results have higher accuracy. After changing the input function and calculus order of the Bagley Torvik equation, the solution results show that the circuit simulation method can solve any nonlinear differential equation.
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