We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.
1. Introduction
In this paper we consider delay differential equations in the reproducing kernel space:
(1)1sxu′′tx+1pxu′hx+1qxumx=gx,IIIIIIIIIIIIIIIIIIIIIIiIIIIIIiiiIIIIIIIIIIIIIIIiIIIIIII0<x<1,u0=A,u1=B,
where u(x)∈W23[0,1] and g(x)∈W21[0,1].
The theory of reproducing kernels [1] was used for the first time at the beginning of the 20th century by S. Zaremba in his work on boundary value problems for harmonic and biharmonic functions. In recent years, a lot of attention has been devoted to the study of RKM to investigate various scientific models. The RKM which accurately computes the series solution is of great interest to applied sciences. The method provides the solution in a rapidly convergent series with components that can be elegantly computed. The book [2] provides excellent overviews of the existing reproducing kernel methods for solving various model problems such as integral and integrodifferential equations.
The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui [3] applied the RKM to handle the second-order boundary value problems. Wang et al. [4] investigated a class of singular boundary value problems by this method and the obtained results were good. Zhou et al. [5] used the RKM effectively to solve second-order boundary value problems. In [6], the method was used to solve nonlinear infinite-delay-differential equations. Wang and Chao [7] and Zhou and Cui [8] independently employed the RKM to variable-coefficient partial differential equations. Geng and Cui [9] and Du and Cui [10] researched the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the RKM. Wu and Li [11] applied iterative reproducing kernel method to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. Yang et al. [12] used this method for solving the system of the linear Volterra integral equations with variable coefficients. A particular singular integral equation was solved by Du and Shen [13]. Barbieri and Meo [14] have studied evaluation of the integral terms in reproducing kernel methods. Third-order three-point boundary value problems were considered by Wu and Li [15]. Chen and Chen [16] investigated the exact solution of system of linear operator equations in reproducing kernel spaces. Akgül has investigated fractional order boundary value problems by RKM [17]. Inc et al. have solved ordinary and partial differential equations by RKM [18–20].
The paper is organized as follows. Section 2 introduces several reproducing kernel spaces. The associated linear operator is presented in Section 3. Section 4 provides the main results. The exact and approximate solutions of problems and an iterative method are developed in the reproducing kernel space in this section. We have proved that the approximate solutions converge to the exact solutions uniformly. Some numerical experiments are illustrated in Section 5. Some conclusions are given in Section 6.
2. Preliminaries2.1. Reproducing Kernel Spaces
In this section, we define some useful reproducing kernel spaces.
Definition 1 (reproducing kernel function).
Let E≠∅. A function K:E×E→C is called areproducing kernel function of the Hilbert space H if and only if
K(·,t)∈H for all t∈E;
φ,K(·,t)=φ(t) for all t∈E and all φ∈H.
The last condition is called “the reproducing property” as the value of the function φ at the point t is reproduced by the inner product of φ with K(·,t).
Definition 2.
We define the space W23[0,1] by
(2)W230,1=u∈AC0,1:u′,u′′∈AC0,1,iiiiiiiiiiiiiiiiiiu(3)∈L20,1,u0=u1=0.
The third derivative of u exists almost everywhere since u′′ is absolutely continuous. The inner product and the norm in W23[0,1] are defined by
(3)u,gW23=∑i=02ui0gi0+∫01u3xg3xdx,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiu,g∈W230,1,uW23=u,uW23,u∈W23[0,1].
The space W23[0,1] is called a reproducing kernel space, as, for each fixed y∈[0,1] and any u∈W23[0,1], there exists a function Ry such that
(4)uy=u,RyW23.
Definition 3.
We define the space W21[0,1] by
(5)W210,1=u∈AC0,1:u′∈L20,1.
The inner product and the norm in W21[0,1] are defined by
(6)u,gW21=∫01uxgx+u′xg′xdx,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiu,g∈G210,1,uW21=u,uW21,u∈W210,1.
The space W21[0,1] is a reproducing kernel space, and its reproducing kernel function Tx is given by Cui and Lin [2]:
(7)Txy=12sinh1coshx+y-1+coshx-y-1.
Lemma 4 (see [<xref ref-type="bibr" rid="B12">21</xref>]).
The space W23[0,1] is a reproducing kernel space, and its reproducing kernel function Ry is given by
(8)Ryx=∑i=16ciyxi-1,x≤y,∑i=16diyxi-1,x>y,
where ci(y) and di(y) coefficients can be found by Maple 16.
3. Solution Representation in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M39"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo mathvariant="bold">[</mml:mo><mml:mn>0,1</mml:mn><mml:mo mathvariant="bold">]</mml:mo></mml:math></inline-formula>
In this section, the solution of (1) is considered in the reproducing kernel space W23[0,1]. On defining the linear operator L:W23[0,1]→W21[0,1] as
(9)Lvx=1sxv′′tx+1pxv′hx+1qxvmx,
model problem (1) takes the form
(10)Lv=f(x,v),x∈0,1,v0=v1=0.
In (9), since v(x) is sufficiently smooth, we see that L:W23[0,1]→W21[0,1] is a bounded linear operator. For convenience, we write u instead of v in (10).
Theorem 5.
The linear operator L defined by (9) is a bounded linear operator.
Proof.
We only need to prove LuW212≤MuW232, where M>0 is a positive constant. By (6), we have
(11)LuW212=Lu,LuW21=∫01Lux2+Lu′x2dx.
By reproducing property, we have
(12)ux=u·,Rx·W23,Lu(x)=u(·),LRx(·)W23,
so
(13)Lu(x)≤uW23LRxW23=M1uW23,
where M1>0 is a positive constant; thus,
(14)∫01Lux2dx≤M12uW232.
Since
(15)Lu′x=u·,LRx′·W23,
we have
(16)Lu′x≤uW23LRx′W23=M2uW23,
where M2>0 is a positive constant, so we have
(17)Lu′t2≤M22uW232,∫01Lu′x2dx≤M22uW232,
that is
(18)LuW212≤∫01Lux2+Lu′x2dxLuW212≤M12+M22uW232=MuW232;
where M=M12+M22>0 is a positive constant. This completes the proof.
4. The Structure of the Solution and the Main Results
From (9), it is clear that L:W23[0,1]→W21[0,1] is a bounded linear operator. Put φi(x)=Txi(x) and ψi(x)=L*φi(x), where L* is conjugate operator of L. The orthonormal system Ψ^i(x)i=1∞ of W23[0,1] can be derived from Gram-Schmidt orthogonalization process of {ψi(x)}i=1∞:
(19)ψ^ix=∑k=1iβikψkx,βii>0,i=1,2,….
Theorem 6.
Let xii=1∞ be dense in [0,1] and ψi(x)=LyRx(y)y=xi. Then the sequence ψi(x)i=1∞ is a complete system in W23[0,1].
Proof.
We have
(20)ψix=L*φix=L*φiy,Rxyψi(x)=φiy,LyRxy=LyRxyy=xi.
The subscript y by the operator L indicates that the operator L applies to the function of y. Clearly, ψi(x)∈W23[0,1]. For each fixed u(x)∈W23[0,1], let u(x),ψi(x)=0,(i=1,2,…), which means that
(21)ux,L*φix=Lu·,φi·=Luxi=0.
Note that xii=1∞ is dense in [0,1]; hence, (Lu)(x)=0. It follows that u≡0 from the existence of L-1. So the proof of Theorem 6 is completed.
Theorem 7.
If u(x) is the exact solution of (10), then
(22)ux=∑i=1∞∑k=1iβikfxk,ukΨ^ix,
where {(xi)}i=1∞ is dense in [0,1].
Proof.
From (19) and uniqueness of solution of (10), we have
(23)ux=∑i=1∞ux,Ψ^ixW23Ψ^ix=∑i=1∞∑k=1iβikux,ΨkxW23Ψ^ix=∑i=1∞∑k=1iβikux,L*φkxW23Ψ^ix=∑i=1∞∑k=1iβikLux,φkxW21Ψ^ix=∑i=1∞∑k=1iβikfx,u,TxkW21Ψ^ix=∑i=1∞∑k=1iβikfxk,ukΨ^ix.
This completes the proof.
Now the approximate solution un(x) can be obtained from the n-term intercept of the exact solution u and
(24)un(x)=∑i=1n∑k=1iβikfxk,ukΨ^ix.
Lemma 8 (see [<xref ref-type="bibr" rid="B10">22</xref>]).
If un-uW23→0, xn→x, (n→∞), and f(x,u) is continuous for x∈[0,1], then
(25)fxn,un-1xn⟶fx,uxasn⟶∞.
Lemma 9 (see [<xref ref-type="bibr" rid="B14">23</xref>]).
For any fixed u0(x)∈W23[0,1], suppose the following conditions are satisfied:
(26)unx=∑i=1nAiψ^ix,(27)Ai=∑k=1iβikfxk,uk-1xk.
unW23 is bounded;
xii=1∞ is dense in [0,1];
f(x,u)∈W21[0,1] for any u(x)∈W23[0,1].
Then un(x) in iterative formula (26) converges to the exact solution of (22) in W23[0,1] and
(28)ux=∑i=1∞Aiψ^ix,
where Ai is given by (27).
We assume that xii=1∞ is dense in [0,1]. Let u(x) be the exact solution of (1) and let un(x) be the n-term approximation solution of (1). We set
(29)uC=maxx∈0,1]ux.
Theorem 10.
If u∈W23[0,1], then
(30)un-uW23⟶0,n⟶∞.
Moreover, a sequence un-uW23 is monotonically decreasing in n.
Proof.
From (22) and (24), it follows that
(31)un-uW23=∑i=n+1∞∑k=1iβikfxk,ukΨ^iW23.
Thus,
(32)un-uW23⟶0,n⟶∞.
In addition,
(33)un-uW232=∑i=n+1∞∑k=1iβikfxk,ukΨ^iW232un-uW232=∑i=n+1∞∑k=1iβikfxk,ukΨ^i2.
Clearly, un-uW23 is monotonically decreasing in n.
Remark 11.
Let us consider countable dense set {x1,x2,…}∈[0,1] and define
(34)φi=Txi,Ψi=L*φi,Ψ^i=∑k=11βikΨk.
Then βik coefficients can be found by
(35)β11=1∥Ψ1∥,βii=1∥Ψi∥2-∑k=1i-1cik2,βij=-∑k=ji-1cikβkj∥Ψi∥2-∑k=1i-1cik2,cik=Ψi,Ψ^k.
4.1. Interpolation for Reproducing Kernel Method
We used interpolation to find the numerical results by RKM with
(36)u(x)=x-bx-ca-ba-cuaiiiiiiiiiii+x-ax-cb-ab-cub+x-ax-bc-bc-auc,
where 0.1≤a<x<b<c≤1. More details for interpolation can be found in [24].
5. Numerical Results
In this section, four numerical examples are provided to show the accuracy of the present method. We used interpolation for Examples 12–14. The RKM does not require discretization of the variables, that is, time and space; it is not effected by computation round-off errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKM for the delay differential equation is controllable and absolute errors are small with present choice of x (see Tables 1–6). The numerical results we obtained justify the advantage of this methodology.
Approximate solutions of Example 12.
x
Approximate solution (m=20)
Approximate solution (m=40)
0.1
−0.14567324629372310303
−0.14332570351241038744
0.2
−0.29985556099058424878
−0.29495827353958461228
0.3
−0.45709584460083115842
−0.44946832894209791578
0.4
−0.61194299763471154735
−0.60142648858052541270
0.5
−0.75894592060247313735
−0.74540337131544229233
0.6
−0.89265351401436369206
−0.875969596007423769
0.7
−1.0076146783806308025
−0.987695781517044783
Approximate solutions of Example 13.
x
Approximate solution (m=20)
Approximate solution (m=40)
1/128
1.007699352586986977
1.0078430972064479777
1/64
1.015467872441933409
1.0157477085866857475
1/32
1.0312145777006698521
1.0317781924005388582
1/16
1.0635619468092503034
1.0646422184208995853
1/8
1.1317728332594306196
1.1336864923985190672
1/4
1.2820670848004034655
1.2853973453552029427
3/8
1.4506420021011940437
1.455402968114665988
Approximate solutions of Example 14.
x
Approximate solution (m=20)
Approximate solution (m=40)
0.4
0.37198086110266406119
0.36562543902764212178
0.6
0.56420118043724686324
0.55433514429791597661
0.7
0.66467348067813109719
0.65290860208735486704
0.8
0.76971776926603696502
0.75593299892958661386
0.9
0.8808862311855404085
0.86493995475233692321
3/8
0.34846550611241626259
0.34252746655728479953
5/8
0.58896615156191184948
0.57863565938605001693
7/8
0.85242546043516774048
0.83703396768442560044
Numerical results of Example 15.
x
Exact solution
Approximate solution (m=20)
Approximate solution (m=40)
0.1
0.001
0.00090776461553418538803
0.00098904754340427120243
0.2
0.008
0.0079254004789810215967
0.0079919839363928962286
0.3
0.027
0.026940356780732013097
0.026994411062098332384
0.4
0.064
0.063953086850948970952
0.063996421220125258843
0.5
0.125
0.12496400548134896886
0.12499809894400479331
0.6
0.216
0.21597349594471182527
0.21599952248650893754
0.7
0.343
0.34298191656657060634
0.343000765213362709
0.8
0.512
0.51198960694198087943
0.51200189692520825458
0.9
0.729
0.72899689388568731882
0.72900298512674395795
1.0
1.000
1.000
1.000
Absolute error for Example 15.
x
Absolute error (m=20)
Absolute error (m=40)
0.1
9.223538446581461197×10-5
1.095245659572879757×10-5
0.2
7.45995210189784033×10-5
8.0160636071037714×10-6
0.3
5.9643219267986903×10-5
5.588937901667616×10-6
0.4
4.6913149051029048×10-5
3.578779874741157×10-6
0.5
3.599451865103114×10-5
1.90105599520669×10-6
0.6
2.650405528817473×10-5
4.7751349106246×10-7
0.7
1.808343342939366×10-5
7.65213362709×10-7
0.8
1.039305801912057×10-5
1.89692520825458×10-6
0.9
3.10611431268118×10-6
2.98512674395795×10-6
1.0
0.0
0.0
Relative error for Example 15.
x
Relative error (m=20)
Relative error (m=40)
0.1
9.223538446581461197×10-2
1.095245659572879757×10-2
0.2
9.3249401273723004125×10-3
1.002007950887971425×10-3
0.3
2.209008121036551963×10-3
2.0699770006176355556×10-4
0.4
7.33017953922328875×10-4
5.5918435542830578125×10-5
0.5
2.8795614920824912×10-4
1.520844796165352×10-5
0.6
1.2270395966747560185×10-4
2.2107106067706481481×10-6
0.7
5.2721380260622915452×10-5
2.2309427484227405248×10-6
0.8
2.0298941443594863281×10-5
3.7049320473722265625×10-6
0.9
4.2607878088905075446×10-6
4.0948240657859396433×10-6
1.0
0.0
0.0
Example 12.
Consider the equation
(37)1p(x)u′(g(x))+1q(x)u(h(x))+1rxux+1sxu′x=Fx,u(0)=0=u(1),
where
(38)gx=x,hx=x3,px=x,qx=x2,rx=x-1,sx=x+1,Fx=expx.
Thus, if the method described above is applied, then we find Table 1.
Example 13.
We take notice of equation
(39)u′′x=ux1+2x2(1+2x)2,u0=1,u1=exp1.
We use transformation
(40)vx=ux-xexp1-1-1
to obtain
(41)v′′x=vx1+2x2+x1+2x2exp1-1+1(1+2x)2v0=0,v1=0.
Thus, if the method described above is applied, then we find Table 2.
Example 14.
We regard the following equation:
(42)u′′x=ux2,u0=0,u1=1.
We use transformation
(43)vx=ux-x
to obtain
(44)v′′x=v(x2)+x2,v0=0,v1=0.
Thus, if the method described above is applied, then we find Table 3.
Example 15.
We consult equation
(45)u′′x=ux+x6-x2,u-1=1,u(1)=1.
We use transformation
(46)vx=ux-x
to obtain
(47)v′′x=vx+x7-x2,v-1=0,v1=0.
The exact solution of (45) is given as
(48)ux=xx2.
Thus, if the method described above is applied, then we find Tables 4, 5, and 6.
6. Conclusion
In this paper, we introduced an algorithm for finding approximate solutions of delay differential equations with RKM. For illustration purposes, four examples were selected to show the computational accuracy. It may be concluded that the RKM is very powerful and efficient in finding approximate solutions for wide classes of problems. Solutions obtained by the present method are uniformly convergent. As shown in Tables 1–6, results of numerical examples show that the present method is an accurate and reliable analytical method for these problems. The present study has confirmed that the RKM offers significant advantages in terms of its straightforward applicability, its computational effectiveness, and its accuracy to solve the strongly nonlinear equations.
Conflict of Interests
The authors declare that they do not have any competing interests or conflict of interests.
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