Multilayer methods are an alternative approach to building the approximate analytical solution of differential equations. This paper presents the study of the results obtained by the implementation of our modifications of acknowledged implicit and explicit numerical methods. The homogeneous Duffing equation is of practical interest for modeling nonlinear oscillations and considered as a model equation. The accuracy of the obtained solutions is compared. It is shown that moving an initial point can significantly increase the accuracy of the solutions.
Modeling the behavior of many real objects often reduces to initial or boundary value
problems for differential equations. In practice, the analytical solution of differential
equations usually cannot be built, therefore, a numerical approximate solution is often
sought. But such solutions are not clear enough. It is complicated to use it for studying
the effect of changing the parameters of the original problem or adjust it to the behavior
of the simulated object using the results of observation. Another well-known approach
is building the approximate analytical solution. A lot of different approaches to finding
it has been developed. There are various asymptotic methods, series expansion
methods, etc. [
Copyright 2021 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
But often the higher-order analytical approximations require analytical solving of
systems of complex nonlinear algebraic equations or additional restrictions on the
parameters or the function [
General Description of Multilayer Methods
The essence of our approach is to apply the well-known recurrence formulas for the
numerical integration of differential equations to an interval with a variable right
endpoint. [
Consider the Cauchy problem for a system of ordinary differential equations (1) (2) (3)
The search for a solution is carried out on the interval [ 0, 0 + ]. According to the main idea of our approach, we use well-known methods for the numerical solution of the problem (1) to an interval with a variable right endpoint ∈ [ 0, 0 + ]. We choose a partition ( ) of the interval [ 0, ] into n subintervals 0 < 1 < ⋯ < < ⋯ < = , ℎ = +1 − . By applying the formula +1 = + ( , ℎ , , +1, , −1, … , 0) ′ = ( , ) { ( 0) = 0
∈ , ∈ n times, we obtain an approximate solution ( ). The operator defines a specific method we modify as described above. The result is a function defined on the interval [ 0, 0 + ]. We can replace an interval [ 0, 0 + ] with [ 0 − , 0 + ]. 3
Multilayer Methods in the Context of Solving the
Duffing Equation
As the model task, we consider the homogeneous Duffing equation with constant
coefficients [
′′ + + 3 = 0 (0) = 0, ′(0) = 1
Higher-order differential equations can always be reduced to the form (1) by increasing the dimension of the system. In our case, it is easy to make a replacement = , ′ = : the formula is used.
3 2 1 1 2 3 y
′ = ′ = −
− 3
(0) = 0, (0) = 1
(4)
The search for a solution is carried out on the interval [
= 1, = 1, 0 = 1, 1 = 1
For simplicity, the partition ( ) is considered uniform for each method, namely ℎ
= ( − 0)⁄ . 3.1
Euler's method. The simplest numerical method for which the operator F has the form ( , ℎ , , ) = ℎ ( , ) Refined Euler's method. Another well-known numerical method in which we used In this case, to start the algorithm, the expression ( , ℎ , , , −1) = 2ℎ ( , ) + −1 − .
1 = 0 + ℎ ( 0 + 1 , 0 +
( 0, 0)) ℎ 1 2 ℎ y x 0.5 1.0 1.5 2.0 2.5 3.0 x ( , ℎ , , ) = ℎ [ ( , ) +
( ′( , ) + ′( , ) ( , ))].
Second-order Runge-Kutta method: ( , ℎ , , ) = ℎ ( , +
( , )) . ℎ 2 y x y x 0.5 x x 3 2 1 1 2 3 y 3 2 1 1 2 3 y 3 2 1 1 2 3 y 3 2 1 1 2 3 y Störmer Method. Since initially, the Duffing equation is a second-order differential equation, we can apply the Störmer method. In this case 2 +1 = 2 − −1 + ℎ ( , ).
This method requires 1. We calculated it approximately by the Taylor formula 1 = 0 + ′( 0) 1! ℎ1 + ′′( 0) 2! ℎ12 + ′′′( 0) 3! ℎ13, where ′′( 0) and ′′′( 0) are easily obtained from differential equation (3). y5 x y x lem (3) and the approximate solution built by our modification of the Störmer Method in the Fig. 6. The plot of the exact solution of problem (3) and the approximate solution built by our modification of the Störmer Method in the case of n=5 case of n=10
Implicit methods are applicable if the equation +1 = + ( , ℎ , , +1, , −1, … , 0) is solvable for +1. About our problem, this means solving the cubic equation at each step. In some cases, it is advisable not to look for the exact solution of this cubic equation but to use the approximate solving methods. We use one step of the Newton method. The justification of this approach is demonstrated below.
The implicit Euler method. The operator F for an implicit method is as follows ( , ℎ , , +1) = ℎ ( +1, +1).
Substituting this expression in (2) we obtain the system { +1 = + ℎ +1 +1 = + ℎ (− +1 − 3+1). (5) This system allows the exact expression +1, +1 in terms of , , ℎ : Then we can use formula to perform computations. The result of such solving is presented below. On the other hand, the solution of system (5) can be obtained by using one step of the Newton method. In this case, we obtain another expression, y5 x y x 0.5 1.0 1.5 2.0 2.5 3.0 x 0.5 1.0 1.5 2.0 2.5 3.0 x 3 3 Fig. 7. The plot of the exact solution of prob- Fig. 8. The plot of the exact solution of problem (3) and the approximate solution built by lem (3) and the approximate solution built by our modification of the implicit Euler method our modification of the implicit Euler method in the case of the exact solution of (5) and n=5 in the case of the approximate solution of (5) (one step of the Newton method) and n=5 The maximum error in the first case is equal to 0.89163, in the second case it is 0.96993. As we can see, there is no significant change in the plot behavior. But when using the Newton method an expression is easier and therefore the complexity and time of calculations are lower than if we solve system (5) analytically. The solutions below are obtained using the Newton method.
One-step Adams method. Another implicit method, the equation has the form
+1 = + ℎ2 ( ( , ) + ( +1, +1)).
y5 x
y x
y x
[0, 1.5]
0.14765
0.022633
0.026272
0.032397
0.0058227
0.16892
0.19285
[
Initial Point Moving To improve the accuracy of the model, we investigated the following approach. Using the methods described above, we build a solution to the Cauchy problem (1) starting from the point 1 ∈ [ 0, 0 + ], other than 0. The unknown initial condition ( 1) = 1 in this case is the parameter of the resulting solution ( , |1). This parameter can be determined from the equation ( 0, |1) = 0. From the computational results below, it follows that by moving an initial point in this way, it is possible to improve the solution on the interval.
As an example, we consider the Störmer method, as the most accurate method of the previously considered. The table below shows the maximum error of the solution obtained by this method on the interval [0,1.5] in the case of moving the initial point from zero in increments of 110. The number of layers n we took equal to 2 and 5. Implicit methods have no advantages over explicit methods for this task. Of the explicit methods we examined, the most accurate is the Störmer method. 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
y 2.0 1.5 1.0 0.5 0.5 1.0 As we can see, solutions with moving the initial point have significantly better accuracy on the interval than a conventional solution built starting from zero. y x 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x x Fig. 11. The plot of the exact solution of problem (3) and the approximate solution built by our modification of the Störmer Method in the case of n=2 and initial point 1 = 0.8 Fig. 32. The plot of the exact solution of problem (3) and the approximate solution built by our modification of the Störmer Method in the case of n=2 and initial point 1 = 0
y 2.0 1.5 1.0 0.5 0.5 1.0 Next, we select new starting points from the gap between the previous best results in increments of 1 . 1/100. From this list of models, the most accurate model can be cho100 sen. Thus, the approach with moving an initial point allowed us to significantly reduce the deviation of the solution on the interval. The plots below illustrate the difference between models with a selected initial point and models built starting from zero. y x 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
y 0.025 0.020 0.015 0.010 0.005 1. The methods we have proposed allow us to construct an approximate solution of the
Duffing equation in the form of a function with the required accuracy.
2. For model equations with parameters considered, implicit methods do not have
significant advantages over explicit methods. Implicit methods make sense for those
parameters when the task becomes stiff.
3. Moving an initial point lets us obtain an approximate analytical solution of the model
task which is several times more accurate than a solution obtained without moving
an initial point. Wherein an accuracy increases with the number of layers.
4. Our methods, without requiring additional assumptions, allow us to build parametric
approximate analytical solutions [
This paper is based on research carried out with the financial support of the grant of the Russian Scientific Foundation (project №18-19-00474).