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# Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching

###### Accettato: 31 Mar 2022
Dettagli della rivista
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Rivista
eISSN
2444-8656
Prima pubblicazione
01 Jan 2016
Frequenza di pubblicazione
2 volte all'anno
Lingue
Inglese
Introduction

Ordinary differential equations' two-end boundary value problem (TPBVP) is frequently encountered in scientific engineering calculations. The classic solution methods include traditional methods such as the “shooting method,” finite difference method, and finite element method [1]. These methods are often unsatisfactory in terms of calculation accuracy and stability. So we first consider the following initial value problem: $v˙=Hv+r(t);v(0)=v0$ \dot v = Hv + r\left( t \right);v\left( 0 \right) = {v_0}

H is the 2 n × 2n dimensional stationary system matrix. r(t) is an arbitrarily varying non-homogeneous term v0 is a given initial condition [2]. The differential equation (1) is a first-order linear non-homogeneous ordinary differential equation about the 2n dimensional state vector v. If the non-homogeneous term r(t) = 0, then the differential equation (1) degenerates into a first-order linear homogeneous ordinary differential equation. Linear means that the system matrix H and the inhomogeneous term r(t) have nothing to do with v, while first-order means that the differential equation has only the first derivative concerning v. The corresponding first-order linear non-homogeneous boundary value problem can be described as $v˙=Hv+r(t);q(0)=q0,q(tf)=qtf$ \dot v = Hv + r\left( t \right);q\left( 0 \right) = {q_0},q\left( {{t_f}} \right) = {q_{{t_f}}} v = (qT pT, tf is the length of the solution area. q0 and qtf are the given boundary conditions at both ends.

Therefore, the problem addressed in this article is the first-order linear non-homogeneous two-end boundary value ordinary differential equation described in equation (2). The solution of the differential equation (1) can be expressed as $v(t)=eHtv0+∫0teH(t−τ)r(τ)dτ$ v\left( t \right) = {e^{Ht}}{v_0} + \int_0^t {{e^{H\left( {t - \tau } \right)}}} r\left( \tau \right)d\tau

In the above formula, the first term at the right end is the general solution of the homogeneous equation corresponding to equation (1), and the second term at the right end is the special solution of the equation (1) corresponding to the non-homogeneous equation. In numerical calculation, we discretize the time domain into a series of equal intervals with a step length of η. Then at any time tk = k η(k = 0,1,2,…) can get the recurrence relationship of the state vector v in a time step by formula (3): $vk+1=Φ0(η)vk+∫0neHτr(tk+1−τ)dτ;Φ0(η)=eHη$ {v_{k + 1}} = {\Phi _0}\left( \eta \right){v_k} + \int_0^n {{e^{H\tau }}} r\left( {{t_{k + 1}} - \tau } \right)d\tau ;{\Phi _0}\left( \eta \right) = {e^{H\eta }}

If the missing n initial states can be obtained by using the n states in the terminal boundary of the boundary value problem at both ends, then the boundary value problem at both ends will be transformed into an initial value problem. In this way, the recursive formula (4) can obtain the state vector vk at any discrete-time tk. The following gives the precise integration calculation process of the homogeneous term state transition matrix Φ0(η) and the non-homogeneous term $∫0ηeHτr(tk+1−τ)dτ$ \int_0^\eta {{e^{H\tau }}} r\left( {{t_{k + 1}} - \tau } \right)d\tau .

Precise integration of the homogeneous general solution Φ0(η)

The precise integration of the general solution Φ0(η) of the homogeneous equation can be calculated using the matrix index method or the section mixed energy method. The matrix index method is simple in principle and easy to program [3]. The zone-mixed energy method is very suitable for solving various problems in modern (optimal) control theory. The following two methods are used to calculate the state transition matrix Φ0(η) of the homogeneous general solution.

Matrix index method

Assume that the area to be solved for the entire boundary value problem at both ends is [x0, xf], and the length is L = xfx0. We discretize L into a series of segments at equal distances, then the segment length is η: $η=L/2M$ \eta = L/{2^M}

Among them M is a positive integer, which controls the size of the discrete step. In the basic section η, we use the fine integration method to calculate the transfer matrix Φ0(η) of the section $Φ0(η)=eHη=(eHη/2N)2N$ {\Phi _0}\left( \eta \right) = {e^{H\eta }} = {\left( {{e^{H\eta /{2^N}}}} \right)^{{2^N}}}

In this way, the state transition matrix Φ0(η) with the section length η is obtained by the standard fine integration method [4]. Further, execute the class 2N algorithm to obtain the state transfer matrix Φ(L) of the entire section: $Φ(L)=eHL=(Φ0(η))2M$ \Phi \left( L \right) = {e^{HL}} = {\left( {{\Phi _0}\left( \eta \right)} \right)^{{2^M}}}

At this time, the terminal state vector can be expressed as the initial state vector as $(qxfpxf)=(Φ11Φ12Φ21Φ22)(qx0px0)$ \left( {\matrix{ {{q_{xf}}} \cr {{p_{xf}}} \cr } } \right) = \left( {\matrix{ {{\Phi _{11}}} & {{\Phi _{12}}} \cr {{\Phi _{21}}} & {{\Phi _{22}}} \cr } } \right)\left( {\matrix{ {{q_{{x_0}}}} \cr {{p_{{x_0}}}} \cr } } \right) $Φ(L)=(Φ11Φ12Φ21Φ22)$ \Phi \left( L \right) = \left( {\matrix{ {{\Phi _{11}}} & {{\Phi _{12}}} \cr {{\Phi _{21}}} & {{\Phi _{22}}} \cr } } \right)

We use equation (8) to connect the initial state and the end state through the state transition matrix Φ (L) of the entire section.

Sectional mixed energy method

By establishing the connection between the state differential equation coefficient matrix and the mixed energy matrix and then solving the differential equation that the mixed energy matrix satisfies, the solution of the asymmetric Riccati differential equation at each time point is obtained [5]. This method can also be used to solve the state transition matrix Φ0(η). The homogeneous differential equation of state is $(qp)=(ADBC)(qp)$ \left( {\matrix{ q \cr p \cr } } \right) = \left( {\matrix{ A & D \cr B & C \cr } } \right)\left( {\matrix{ q \cr p \cr } } \right) q is the n dimensional state vector. p is the m dimensional state vector. A, D, B, C are the coefficient matrix with the corresponding dimension. We use the section mixed energy method to describe. The state at both ends of the section is described by the mixed energy matrix as $(qbpa)=(F−GQE)(qapb)$ \left( {\matrix{ {{q_b}} \cr {{p_a}} \cr } } \right) = \left( {\matrix{ F & { - G} \cr Q & E \cr } } \right)\left( {\matrix{ {{q_a}} \cr {{p_b}} \cr } } \right) qa, pa are the state vector at the a end of the section. qb, pb is the state vector at the b end of the section. F, G, Q, E are the mixed energy matrix of this section. The section mixing energy matrix also has section merging formulas [6]. Assume that section 1:[ xa, xb] and section 2:[ xb, xc] are as shown in Figure 1. The segment mixing energy matrix is Q1, G1, F1, E1, and Q2, G2, F2, E2. We combine the two sections into a whole. The zone mixing energy matrix Qc, Gc, Fc, Ec corresponding to zone 3[xa, xc] is shown in the following formula.

Sections 1 and 2 are merged into the whole section 3, and the section mixed energy merge formula: $Qc=Q1+E1(Im+Q2G1)−1Q2F1Gc=G2+F2(In+G1Q2)−1G1E2Fc=F2(In+G1Q2)−1F1Ec=E1(Im+Q2G1)−1E2$ \matrix{ {{Q_c}} \hfill & = \hfill & {{Q_1} + {E_1}{{\left( {{I_m} + {Q_2}{G_1}} \right)}^{ - 1}}{Q_2}{F_1}} \hfill \cr {{G_c}} \hfill & = \hfill & {{G_2} + {F_2}{{\left( {{I_n} + {G_1}{Q_2}} \right)}^{ - 1}}{G_1}{E_2}} \hfill \cr {{F_c}} \hfill & = \hfill & {{F_2}{{\left( {{I_n} + {G_1}{Q_2}} \right)}^{ - 1}}{F_1}} \hfill \cr {{E_c}} \hfill & = \hfill & {{E_1}{{\left( {{I_m} + {Q_2}{G_1}} \right)}^{ - 1}}{E_2}} \hfill \cr }

We use type 2N algorithms in the section of length η. The article divides the section of length η into smaller sections of length τ = η/2N. At the same time, we expand the segment mixing matrix series in this very small segment τ. According to the equation satisfied by the section mixing energy matrix, the connection between the section mixing energy matrix and the coefficient matrix of the homogeneous state differential equation (9) is established to obtain the analytical expression of the section mixing energy matrix in a very small section τ. The mixed energy matrix Q(η), G(η), F(η), E(η) with the section length η is obtained by the section merging formula. Then execute the mixed energy combination formula (11) of the whole section L to obtain the mixed energy matrix Q(L), G(L), F(L), E(L), of the whole section [7]. Therefore, the two states are connected, as shown in equation (10). We express it in the form of (8) transfer matrix $Φ11=F(L)+G(L)E−1(L)Q(L)Φ12=−G(L)E−1(L)Φ21=−E−1(L)Q(L)Φ22=−E−1(L)$ \matrix{ {{\Phi _{11}}} \hfill & = \hfill & {F\left( L \right) + G\left( L \right){E^{ - 1}}\left( L \right)Q\left( L \right)} \hfill \cr {{\Phi _{12}}} \hfill & = \hfill & { - G\left( L \right){E^{ - 1}}\left( L \right)} \hfill \cr {{\Phi _{21}}} \hfill & = \hfill & { - {E^{ - 1}}\left( L \right)Q\left( L \right)} \hfill \cr {{\Phi _{22}}} \hfill & = \hfill & { - {E^{ - 1}}\left( L \right)} \hfill \cr }

Equation (12) is the corresponding relationship between the state transition matrix Φ (L) and the mixed energy matrix Q(L), G(L), F(L), E(L) in the section L.

Fine integration of inhomogeneous terms

We use the 2N type algorithm to expand the approximate series of the inhomogeneous terms in a very small section τ. Then use the method of section merging to obtain the expression of the inhomogeneous terms in the introductory section η. The following gives specific derivations for inhomogeneous terms as polynomials, trigonometric functions, and exponential functions [8]. If the non-homogeneous terms are in polynomial form, for example, the following linear non-homogeneous terms $r(x)=rk+r1(x−xk);r1=(rk+1−rk)/η$ r\left( x \right) = {r_k} + {r_1}\left( {x - {x_k}} \right);{r_1} = \left( {{r_{k + 1}} - {r_k}} \right)/\eta

Then the corresponding inhomogeneous term in equation (4) is specifically solved as $F(η)=Φ1(η)rk+1−Φ2(η)(rk+1−rk)/η$ F\left( \eta \right) = {\Phi _1}\left( \eta \right){r_{k + 1}} - {\Phi _2}\left( \eta \right)\left( {{r_{k + 1}} - {r_k}} \right)/\eta

In $Φ1(η)=∫0ηeHτdτ; Φ2(η)=∫0ηeHττdτ$ {\Phi _1}\left( \eta \right) = \int_0^\eta {{e^{H\tau }}} d\tau ;\,{\Phi _2}\left( \eta \right) = \int_0^\eta {{e^{H\tau }}} \tau d\tau

After the partial integration operation, Φ1(η), Φ2(η) itself has the section merging formula expressed as follows $Φ1(2η)=Φ1(η)+Φ0(η)Φ1(η);Φ2(η)=Φ2(η)+Φ0(η)(ηΦ1(η)+Φ2(η))$ \matrix{ {{\Phi _1}\left( {2\eta } \right) = {\Phi _1}\left( \eta \right) + {\Phi _0}\left( \eta \right){\Phi _1}\left( \eta \right);} \hfill \cr {{\Phi _2}\left( \eta \right) = {\Phi _2}\left( \eta \right) + {\Phi _0}\left( \eta \right)\left( {\eta {\Phi _1}\left( \eta \right) + {\Phi _2}\left( \eta \right)} \right)} \hfill \cr }

The section matrix Φ1 (L), Φ2 (L) of the entire section L is obtained by executing the 2N typing algorithm. This process is expressed in programming language as

for (i =0; i < M; i + +)

{

Φ1 = Φ1 + Φ0Φ1;

Φ2 = Φ2 + Φ0 (ηΦ1 + Φ2);

η = η + η;

}

After we get Φ1 (L), Φ2 (L), the inhomogeneous term of the entire section L can be expressed as $F(L)=Φ1(L)rxf−Φ2(L)(rxf−rx0)/L$ F\left( L \right) = {\Phi _1}\left( L \right){r_{{x_f}}} - {\Phi _2}\left( L \right)\left( {{r_{{x_f}}} - {r_{{x_0}}}} \right)/L

If the inhomogeneous term is a sine/cosine function, there is an expression $r(x)=r1sin(γx)+r2cos(γx)$ r\left( x \right) = {r_1}\sin \left( {\gamma x} \right) + {r_2}\cos \left( {\gamma x} \right)

Then the special solution corresponding to the inhomogeneous term is $F(η)=Ec(η)ak+1+Es(η)bk+1$ F\left( \eta \right) = {E_c}\left( \eta \right){a_{k + 1}} + {E_s}\left( \eta \right){b_{k + 1}}

In $Ec(η)=∫0ηcos (γτ)eHτdτEs(η)=∫0ηsin (γτ)eHτdτ$ \matrix{ {{E_c}\left( \eta \right) = \int_0^\eta {\cos \,\left( {\gamma \tau } \right){e^{H\tau }}d\tau } } \hfill \cr {{E_s}\left( \eta \right) = \int_0^\eta {\sin \,\left( {\gamma \tau } \right){e^{H\tau }}d\tau } } \hfill \cr } $ak+1=r1sin(γxk+1)+r2cos(γxk+1)bk+1=−r1cos(γxk+1)+r2sin(γxk+1)$ \matrix{ {{a_{k + 1}} = {r_1}\sin \left( {\gamma {x_{k + 1}}} \right) + {r_2}\cos \left( {\gamma {x_{k + 1}}} \right)} \hfill \cr {{b_{k + 1}} = - {r_1}\cos \left( {\gamma {x_{k + 1}}} \right) + {r_2}\sin \left( {\gamma {x_{k + 1}}} \right)} \hfill \cr }

Among them, Ec(η), Es(η), can also be calculated by fine integration. Similarly, there is the following section merging formulas through the partial integration operation: $Es(2η)=Es(η)+Φ0(η)(cos(γη)Es(η)+sin(γη)Ec(η))Ec(2η)=Ec(η)+Φ0(η)(cos(γη)Ec(η)−sin(γη)Es(η))$ \matrix{ {{E_s}\left( {2\eta } \right)} & = & {{E_s}\left( \eta \right) + {\Phi _0}\left( \eta \right)\left( {\cos \left( {\gamma \eta } \right){E_s}\left( \eta \right) + \sin \left( {\gamma \eta } \right){E_c}\left( \eta \right)} \right)} \cr {{E_c}\left( {2\eta } \right)} & = & {{E_c}\left( \eta \right) + {\Phi _0}\left( \eta \right)\left( {\cos \left( {\gamma \eta } \right){E_c}\left( \eta \right) - \sin \left( {\gamma \eta } \right){E_s}\left( \eta \right)} \right)} \cr }

We use the 2N typing algorithm to get the Ec(L), Es(L) of the entire section L, which is expressed in programming language as

for (i =0; i < M; i + +)

{

Es = Es + Φ0 (cos(γη) Es + sin(γη) (γη) Ec);

Ec = Ec + Φ0 (cos(γη) Ec + sin(γη) (γη) Es);

Φ0 = Φ0 Φ0;

η = η + η;

}

After obtaining the matrix Ec (L), Es (L), the inhomogeneous term of the entire section L can be expressed as $F(L)=Ec(L)axf+Es(L)Bxf$ F\left( L \right) = {E_c}\left( L \right){a_{{x_f}}} + {E_s}\left( L \right){B_{{x_f}}}

If the inhomogeneous term is an exponential function, the expression is as follows $r(x)=eaxf$ r\left( x \right) = {e^{ax}}f

Then the special solution corresponding to the inhomogeneous term is $F(η)=eaxk+1 Ee(η) f$ F\left( \eta \right) = {e^{a{x_{k + 1}}}}\,{E_e}\left( \eta \right)\,f

In $Ee(η)=∫0ηeHτe−aτdτ$ {E_e}\left( \eta \right) = \int_0^\eta {{e^{H\tau }}} {e^{ - a\tau }}d\tau

After the division integration calculation, the following section merging formula is available: $Ee(2η)=Ee(η)+e−aηΦ0(η)Ee(η)$ {E_e}\left( {2\eta } \right) = {E_e}\left( \eta \right) + {e^{ - a\eta }}{\Phi _0}\left( \eta \right){E_e}\left( \eta \right)

The Ee(L), Es(L), of the entire section, L is obtained by executing the typing 2N algorithm, which is expressed in programming language as

for (i =0; i < M; i + +)

{

Ee = Ee + e Φ0Ee;

Φ0 = Φ0 Φ0;

η = η + η;

After executing the above program statement, the inhomogeneous term expression of the entire section L is obtained: $F(L)=eaxfEe(L) f$ F\left( L \right) = {e^{a{x_f}}}{E_e}\left( L \right)\,f

When the inhomogeneous term is the above function itself or a combination between them, the numerical result calculated by fine integration is almost the exact solution on the computer. When the inhomogeneous terms are other functions, it can be expressed in section η by fitting polynomials, Fourier series expansion, and other methods into the above functional form [9]. At this time, the calculation accuracy of the inhomogeneous term mainly depends on the accuracy of this approximation.

Precise integration solution for the boundary value problem at both ends of the linear non-homogeneous equation

With the above precise integration process of the homogeneous term state transition matrix Φ (L) and the non-homogeneous term F(L), we can consider the precise integration method for the boundary value problem at both ends of the linear non-homogeneous equation [10]. The following equation can connect the boundary value problem of the state vector at both ends $(qxfpxf)=Φ(L)(qx0px0)+F(L)$ \left( {\matrix{ {{q_{{x_f}}}} \cr {{p_{{x_f}}}} \cr } } \right) = \Phi \left( L \right)\left( {\matrix{ {{q_{{x_0}}}} \cr {{p_{{x_0}}}} \cr } } \right) + F\left( L \right)

In $Φ(L)=(Φ11Φ12Φ21Φ22), F(L)(FqFp)$ \Phi \left( L \right) = \left( {\matrix{ {{\Phi _{11}}} & {{\Phi _{12}}} \cr {{\Phi _{21}}} & {{\Phi _{22}}} \cr } } \right),\,F\left( L \right)\left( {\matrix{ {{F_q}} \cr {{F_p}} \cr } } \right)

If the boundary conditions at both ends are known, such as qx0 and qxf, the state vector px0 with an unknown initial end can be obtained: $px0=Φ12−1(qxf−Φ11qx0−Fq)$ {p_{{x_0}}} = \Phi _{12}^{ - 1}\left( {{q_{{x_f}}} - {\Phi _{11}}{q_{{x_0}}} - {F_q}} \right)

In this way, the complete initial conditions qx0 and px0, are obtained. At this time, it can be solved completely according to the initial value problem [11]. When the sections are merged, the state transition matrix Φ0(η) and the inhomogeneous term F(η) of section η can be stored. In this way, the state vector value vk at the endpoint k of each section can be solved according to the precise integration recursive format of the initial value problem.

Numerical examples

Solve the boundary value problem of a second-order ordinary differential equation $y″(x)+14y(x)=8$ y''\left( x \right) + {1 \over 4}y\left( x \right) = 8 . The boundary condition at both ends is y(0) = 0, y(10) = 0. The analytical solution is $y(x)=32(cos 5−2sin 5sinx2−cosx2+1)$ y\left( x \right) = 32\left( {{{\cos \,5 - 2} \over {\sin \,5}}\sin {x \over 2} - \cos {x \over 2} + 1} \right)

Select M = 5 as the step parameter. The absolute error accuracy e calculated by the method in this paper is shown in Figure 2. It can be seen from Fig. 2 that the absolute error accuracy of the calculation result is in 10−13 – 10−16 regardless of whether the matrix index method or the Riccati method is used. Both have quite high calculation accuracy and almost reach the exact solution on the computer.

Example 2: We solve a boundary value problem at both ends of a stiff ordinary differential equation. The differential equation is $(du/dxdυ/dx)=(9981998−999−1998) (uυ)+(11)(1+x)e−x$ \left( {\matrix{ {du/dx} \cr {d\upsilon /dx} \cr } } \right) = \left( {\matrix{ {998} & {1998} \cr { - 999} & { - 1998} \cr } } \right)\,\left( {\matrix{ u \cr \upsilon \cr } } \right) + \left( {\matrix{ 1 \cr 1 \cr } } \right)\left( {1 + x} \right){e^{ - x}}

The eigenvalues of the system matrix H are λ1 = −1, λ2 = −1000 respectively, and the difference is relatively large. It has obvious rigid characteristics. The given boundary conditions are $u(0)=1, υ(1)=331 669332 667e−1000−1328 671332 667e−1$ u\left( 0 \right) = 1,\,\,\upsilon \left( 1 \right) = {{331\,669} \over {332\,667}}{e^{ - 1000}} - {{1328\,671} \over {332\,667}}{e^{ - 1}}

At this time, the analytical solution is $u*(x)=−331 669332 667e−1000x+664 336332 667e−x+1331333e−xx+2e−xx2υ*(x)=−331 669332 667e−1000x−331 669332 667e−x−665333e−xx−e−xx2$ \matrix{ {{u^*}\left( x \right)} \hfill & = \hfill & { - {{331\,669} \over {332\,667}}{e^{ - 1000x}} + {{664\,336} \over {332\,667}}{e^{ - x}} + {{1331} \over {333}}{e^{ - x}}x + 2{e^{ - x}}{x^2}} \hfill \cr {{\upsilon ^*}\left( x \right)} \hfill & = \hfill & { - {{331\,669} \over {332\,667}}{e^{ - 1000x}} - {{331\,669} \over {332\,667}}{e^{ - x}} - {{665} \over {333}}{e^{ - x}}x - {e^{ - x}}{x^2}} \hfill \cr }

For this rigid problem, we use the method of this paper to calculate. Select = 5 M = 5 for the long parameter, and calculate the absolute error as shown in Figure 3. It can be seen from the figure that the calculation accuracy of this method is also in the 10−10 – 10−16 order. The Riccati method has a slightly lower calculation accuracy than the matrix index method. Because in the process of calculating the zone mixing energy matrix and then transforming it into the state transition matrix, it is necessary to invert part of the matrix. This will cause numerical errors in the inversion process for severe problems.

Conclusion

This paper adopts the precise integration method to calculate the state transition matrix and non-homogeneous term of the homogeneous general solution. At the same time, we transformed the boundary value problem at both ends of the linear non-homogeneous ordinary differential equation into the initial value problem. We introduced the recursive form of the precise integration solution process. The method in this paper only needs to perform a precise integration operation on the system matrix once. The article does not need to invert the system matrix because there is no approximation of the different classification algorithms for inhomogeneous terms. Numerical examples show that the algorithm has high reliability and accuracy.

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