Numerical Solution of Seventh-Order Boundary Value Problems by a Novel Method

Abstract

We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventh-order boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), variation of parameters method (VPM), and homotopy analysis method (HAM). Obtained results show that our method is very effective.

1. Introduction

Consider the seventh-order boundary value problem [1–5]: 𝑢⁽⁷⁾(𝑥)=𝑁(𝑥,𝑢(𝑥)),0≤𝑥≤1, (1) with boundary conditions 𝑢^(𝑖)(0)=𝐴_𝑖𝑢,𝑖=0,1,2,3,_(𝑗)(1)=𝐵_𝑗,𝑗=0,1,2. (2) The analytical solution of seventh-order differential equations are rarely exists in literature. However, there are various numerical methods for the solution of (1)-(2). The aim of this work is to apply reproducing kernel Hilbert space method (RKHSM) [6–28] to solve the seventh-order boundary value problems. Numerical results of the seventh-order boundary value problems have been obtained by this method in our work. This study shows that the proposed method can be considered as an alternative technique for solving linear and nonlinear problems in science and engineering [29–31].

The paper is organized as follows. Section 2 introduces several reproducing kernel spaces. We provide the main results and the exact and approximate solutions of (1)-(2) in Section 3. We have proved that the approximate solution converges to the exact solution uniformly. Some numerical experiments are illustrated in Section 4. There are some conclusions in the last section.

2. Reproducing Kernel Spaces

In this section, we define some useful reproducing kernel spaces.

Definition 1. We define the space 𝐻¹₂[0,1] by 𝐻¹₂[]=[],𝑓0,1𝑓∣𝑓isabsolutelycontinuousin0,1_′(𝑥)∈𝐿²[0,1],𝑥∈[0,1]. (3) The inner product and the norm in 𝐻¹₂[0,1] are defined, respectively, by 𝑓,𝑔_𝐻12=𝑓(0)𝑔(0)+¹₀𝑓^′(𝑥)𝑔^′(𝑥)𝑑𝑥,𝑢,𝑓∈𝐻¹₂[],𝑓0,1_𝐻12=𝑓,𝑓_𝐻12,𝑓∈𝐻¹₂[0,1]. (4) The space 𝐻¹₂[0,1] is a reproducing kernel space and its reproducing kernel function 𝑟_𝑥 is given by 𝑟_𝑥𝑦=1+𝑥,𝑥≤𝑦,1+𝑦,𝑥>𝑦. (5)

Definition 2. We define the space 𝑇⁸₂[0,1] by the following: 𝑇⁸₂[0,1]=𝑓∣𝑓,𝑓^′,𝑓^′′,𝑓⁽³⁾,𝑓⁽⁴⁾,𝑓⁽⁵⁾,𝑓⁽⁶⁾,𝑓⁽⁷⁾[],𝑓areabsolutelycontinuousin0,1₍₈₎∈𝐿²[][],𝑓(0)0,1,𝑥∈0,1=𝑓^′(0)=𝑓^′′(0)=𝑓⁽³⁾(0)=𝑓(1)=𝑓^′(1)=𝑓^′′(1)=0. (6) The inner product and the norm in 𝑇⁸₂[0,1] are defined, respectively, by 𝑓,𝑔_𝑇82=⁷_𝑖=0𝑓^(𝑖)(0)𝑔^(𝑖)(0)+¹₀𝑓⁽⁸⁾(𝑥)𝑔⁽⁸⁾(𝑥)𝑑𝑥,𝑓,𝑔∈𝑇⁸₂[],𝑓0,1_𝑇82=𝑓,𝑓_𝑇82,𝑓∈𝑇⁸₂[0,1]. (7) The space 𝑇⁸₂[0,1] is a reproducing kernel space; that is, for each fixed 𝑦∈[0,1] and any 𝑓∈𝑇⁸₂[0,1], there exists a function 𝑅_𝑦 such that 𝑓=𝑓,𝑅_𝑦𝑇82. (8)

Theorem 3. The space 𝑇⁸₂[0,1] is a reproducing kernel Hilbert space whose reproducing kernel function 𝑅_𝑦 is given by 𝑅_𝑦(𝑥)=^16𝑖=1𝑐_𝑖𝑦𝑥^𝑖−1,𝑥≤𝑦,_16𝑖=1𝑑_𝑖𝑦𝑥^𝑖−1,𝑥>𝑦, (9) where 𝑐_𝑖(𝑦) and 𝑑_𝑖(𝑦) can be obtained by Maple 16 and proof of Theorem 3 is given in Appendix.

3. Exact and Approximate Solutions of (1)-(2) in 𝑇⁸₂[0,1]

The solution of (1)-(2) is given in the reproducing kernel space 𝑇⁸₂[0,1]. The linear operator 𝐿∶𝑇⁸₂[0,1]⟶𝐻¹₂[0,1] (10) is bounded. After homogenizing the boundary conditions, we obtain ((𝑥𝑣𝐿𝑣=𝑀𝑥,𝑣)),0≤𝑥≤1,_(𝑖)(0)𝑣=0,𝑖=0,1,2,3,_(𝑗)(1)=0,𝑗=0,1,2. (11) We choose a countable dense subset 𝑃={𝑥_𝑖}^∞_𝑖=1 in [0,1] and let Ψ_𝑥𝑦=𝐿^∗𝑟_𝑥𝑦, (12) where 𝐿^∗ is conjugate operator of 𝐿 and 𝑟_𝑥 is given by (5). Furthermore, for simplicity let Ψ_𝑖(𝑥)=Ψ_𝑥𝑖(𝑥); namely, Ψ_𝑖(𝑥)=^defΨ_𝑥𝑖(𝑥)=𝐿^∗𝑟_𝑥𝑖(𝑥). (13) Now one can deduce the following lemmas.

Lemma 4. {Ψ_𝑖(𝑥)}^∞_𝑖=1 is complete system of 𝑇⁸₂[0,1].

Proof. For 𝑓∈𝑇⁸₂[0,1], let ⟨𝑓,Ψ_𝑖⟩=0(𝑖=1,2,…); that is, 𝑓,𝐿^∗𝑟_𝑥𝑖=𝐿𝑓𝑥_𝑖=0. (14) Note that {𝑥_𝑖}^∞_𝑖=1 is the dense set in [0,1]; therefore, (𝐿𝑓)(𝑥)=0. It follows that 𝑓(𝑥)=0 from the existence of 𝐿⁻¹.

Lemma 5. The following formula holds: Ψ_𝑖(𝑥)=𝐿_𝜈𝑅_𝑥(𝜈)𝑥_𝑖, (15) where the subscript 𝜈 of operator 𝐿_𝜈 indicates that the operator 𝐿 applies to function of 𝜈.

Proof. Consider the following: Ψ_𝑖(𝑥)=Ψ_𝑖(𝜉),𝑅_𝑥(𝜉)_𝑇82=𝐿^∗𝑟_𝑥𝑖(𝜉),𝑅_𝑥(𝜉)_𝑇82=𝑟_𝑥𝑖(𝜉),𝐿_𝜈𝑅_𝑥(𝜈)(𝜉)_𝐻12=𝐿_𝜈𝑅_𝑥(𝜈)𝑥_𝑖. (16) This completes the proof.

Remark 6. The orthonormal system {Ψ_𝑖(𝑥)}^∞_𝑖=1 of 𝑇⁸₂[0,1] can be derived from Gram-Schmidt orthogonalization process of {Ψ_𝑖(𝑥)}^∞_𝑖=1, Ψ_𝑖(𝑥)=^𝑖_𝑘=1𝛽_𝑖𝑘Ψ_𝑘(𝑥),𝛽_𝑖𝑖>0,𝑖=1,2,…, (17) where 𝛽_𝑖𝑘 are orthogonal coefficients.

In the following, we will give the representation of the exact solution of (11) in the reproducing kernel space 𝑇⁸₂[0,1].

Theorem 7. If 𝑢 is the exact solution of (11), then 𝑢=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑀𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥), (18) where {𝑥_𝑖}^∞_𝑖=1 is a dense set in [0,1].

Proof. From the (17) and uniqueness of solution of (11), we have 𝑢=^∞_𝑖=1𝑢,Ψ_𝑖𝑇82Ψ_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢,𝐿^∗𝑟_{𝑥𝑘𝑇82}Ψ_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝐿𝑢,𝑟_{𝑥𝑘𝐻12}Ψ_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑀,𝑟_{𝑥𝑘𝐻12}Ψ_𝑖=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑀𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥). (19) This completes the proof.

Now the approximate solution 𝑢_𝑛 can be obtained by truncating the 𝑛-term of the exact solution 𝑢 as 𝑢_𝑛=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑀𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥). (20)

Lemma 8. Assume 𝑢 is the solution of (11) and 𝑟_𝑛 is the error between the approximate solution 𝑢_𝑛 and the exact solution 𝑢. Then the error sequence 𝑟_𝑛 is monotone decreasing in the sense of ‖⋅‖_𝑇⁸₂ and ‖𝑟_𝑛(𝑥)‖_𝑇⁸₂→0.

Proof. From (18) and (20), we obtain 𝑢−𝑢_𝑛𝑇82=^∞𝑖_{𝑖=𝑛+1𝑘=1}𝛽_𝑖𝑘𝑀𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥)_𝑇82. (21) Thus 𝑢−𝑢_𝑛𝑇82⟶0,𝑛⟶∞. (22) In addition 𝑢−𝑢_𝑛²_𝑇82=^∞𝑖_{𝑖=𝑛+1𝑘=1}𝛽_𝑖𝑘𝑀𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥)²_𝑇82=^∞_𝑖=𝑛+1^𝑖_𝑘=1𝛽_𝑖𝑘𝑀𝑥_𝑘,𝑢𝑥_𝑘Ψ_𝑖(𝑥)_𝑇⁸₂². (23) Then, ‖𝑢−𝑢_𝑛‖_𝑇⁸₂ is monotonically decreasing in 𝑛.

Remark 9. The seventh-order boundary value problems have come out in construction engineering, beam column theory, and chemical reactions. Therefore solutions of the seventh-order boundary value problems are very important in the literature. The reproducing kernel function for seventh-order boundary value problem has not been calculated till now. All computations are performed by Maple 16. The RKHSM does not require discretization of the variables, that is, time and space, and it is not affected by computational round-off errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKHSM for the seventh-order boundary value problems is controllable and absolute errors are small with present choice of 𝑥 (see Tables 1–6 and Figures 1–6). The obtained numerical results justify the advantage of this methodology. We gave transformations to homogenize the boundary conditions for all examples. Additionally, we improved our programme to find numerical results. As shown in Tables 1, 3, and 5 all the numerical results have been found in very short time.

Table 1 

Numerical results for Example 10 (time (s): 1.645).

Table 2 

Comparison of absolute error of HPM, VIM, and RKHSM for Example 10.

Table 3 

Numerical results for Example 11 (time (s): 3.123).

Table 4 

Comparison of absolute error of VPM, ADM, HAM, and RKHSM.

Table 5 

Numerical results for Example 12 (time (s): 5.234).

Table 6 

Comparison of absolute error of ADM, HAM, and RKHSM.

Figure 1 

Comparison of analytical solution and RKHSM solution for Example 10.

Figure 2 

Comparison of absolute error of VIM, HPM, and RKHSM for Example 10.

Figure 3 

Comparison of analytical solution and RKHSM solution for Example 11.

Figure 4 

Comparison of absolute error of ADM, VPM, HAM, and RKHSM for Example 11.

Figure 5 

Comparison of analytical solution and RKHSM solution for Example 12.

Figure 6 

Comparison of absolute error of ADM, HAM, and RKHSM for Example 12.

4. Numerical Results

In this section, three numerical examples are provided to show the accuracy of the present method.

Example 10. We first consider the seventh-order nonlinear boundary value problem: 𝑢⁽⁷⁾(𝑥)=𝑒^−𝑥𝑢²(𝑥)𝑢(0),0<𝑥<1,=𝑢^′(0)=𝑢^′′(0)=𝑢⁽³⁾(0)𝑢(1)=1,=𝑢^′(1)=𝑢^′′(1)=𝑒. (24) The exact solution of (24) is given as [1] 𝑢(𝑥)=𝑒^𝑥. (25)
After homogenizing the boundary conditions of (24), we obtain 𝑣⁽⁷⁾(𝑥)=𝑒^−𝑥𝑣(𝑥)𝑥+1+𝑥+²2+𝑥³6+𝑥⁴212𝑒−572+𝑥⁵872−16𝑒+𝑥⁶132𝑒−533²,𝑣(0)0≤𝑥≤1,=𝑣^′(0)=𝑣^′′(0)=𝑣⁽³⁾(0)𝑣(1)=0,=𝑣^′(1)=𝑣^′′(1)=0, (26) where we used the following transformation: 𝑣(𝑥)(𝑥)𝑥=𝑢−1−𝑥−²2−𝑥³6−𝑥⁴212𝑒−572−𝑥⁵872−16𝑒−𝑥⁶132𝑒−533. (27) Using the RKHSM for this example we obtain Tables 1-2 and Figures 1-2.

Example 11. We now consider the seventh-order linear BVP 𝑢⁽⁷⁾(𝑥)=−𝑢(𝑥)−𝑒^𝑥35+12𝑥+2𝑥²𝑢(0),0≤𝑥≤1,=0,𝑢^′(0)𝑢=1,_′′(0)=0,𝑢⁽³⁾(0)𝑢(1)=−3,=0,𝑢^′(1)=−𝑒,𝑢^′′(1)=−4𝑒. (28) The exact solution of (28) is given as [3] 𝑢(𝑥)=𝑥(1−𝑥)𝑒^𝑥. (29)
After homogenizing the boundary conditions of (28), we get 𝑣⁽⁷⁾(𝑥)+𝑣(𝑥)=−𝑒^𝑥35+12𝑥+2𝑥²+𝑥−𝑥₃2−𝑥⁴3𝑒−172−𝑥⁵272−5𝑒−𝑥⁶2𝑒−112𝑣(0),0≤𝑥≤1,=𝑣^′(0)=𝑣^′′(0)=𝑣⁽³⁾(0)𝑣(1)=0,=𝑣^′(1)=𝑣^′′(1)=0, (30) where, we used the following transformation 𝑣(𝑥)=𝑢(𝑥)−𝑥+𝑥^3/2−𝑥⁴3𝑒−172−𝑥⁵272−5𝑒−𝑥⁶2𝑒−112. (31) Using RKHSM for this example we obtain Tables 3-4 and Figures 3-4.

Example 12. Consider the following seventh-order nonlinear BVP 𝑢⁽⁷⁾(𝑥)=𝑢(𝑥)𝑢^′(𝑥)+𝑒^−2𝑥2+𝑒^𝑥(𝑥−8)−3𝑥+𝑥²,𝑢(0)0≤𝑥≤1,=1,𝑢^′(0)𝑢=−2,_′′(0)=3,𝑢⁽³⁾(0)𝑢(1)=−4,=0,𝑢^′(1)=−1𝑒,𝑢^′′(1)=2𝑒. (32) The exact solution of (32) is given as [3] 𝑢(𝑥)=(1−𝑥)𝑒^−𝑥. (33)
After homogenizing the boundary conditions of (32), we have𝑣⁽⁷⁾(𝑥)−−2+3𝑥−2𝑥²+4𝑥³6𝑒−2+5𝑥⁴4−11𝑒+6𝑥⁵5𝑒−116𝑣(𝑥)−31−2𝑥+2𝑥²−23𝑥³+𝑥⁴6𝑒−2+𝑥⁵4−11𝑒+𝑥⁶5𝑒−116𝑣^′(𝑥)=31−2𝑥+2𝑥²−23𝑥³+𝑥⁴6𝑒−2+𝑥⁵4−11𝑒+𝑥⁶5𝑒−116−2+3𝑥−2𝑥²+4𝑥³6𝑒−2+5𝑥⁴4−11𝑒+6𝑥⁵5𝑒−116+𝑣(𝑥)𝑣^′(𝑥)+𝑒^−2𝑥2+𝑒^𝑥(𝑥−8)−3𝑥+𝑥²𝑣(0),0≤𝑥≤1,=𝑣^′(0)=𝑣^′′(0)=𝑣⁽³⁾(0)𝑣(1)=0,=𝑣^′(1)=𝑣^′′(1)=0, (34)

where we used the following transformation: 𝑢(𝑥)(𝑥)3=𝑣+1−2𝑥+2𝑥²−23𝑥³+𝑥⁴6𝑒−2+𝑥⁵4−11𝑒+𝑥⁶5𝑒−116. (35)

Using RKHSM for this example we obtain Tables 5-6 and Figures 1–6.

Remark 13. Using our method we chose 36 points on [0,1]. In Tables 1–6, we computed the absolute errors |𝑢(𝑥,𝑡)−𝑢_𝑛(𝑥,𝑡)| at the points {(𝑥_𝑖)∶𝑥_𝑖=𝑖,𝑖=0.0,0.1,…,1.0}. The RKHSM tested on three problems, one linear and two nonlinear. A comparison with VIM [1], HPM [2], ADM [3], VPM [4], and HAM [5] was made and it was seen that the present method yields good results (see Tables 1–6 and Figures 1–6).

5. Conclusion

In this paper, we introduced an algorithm for solving the seventh-order problem with boundary conditions. For illustration purposes, we chose three examples which were selected to show the computational accuracy. It may be concluded that the RKHSM is very powerful and efficient in finding exact solution for a wide class of boundary value problems. The approximate solution obtained by the present method is uniformly convergent. Clearly, the series solution methodology can be applied to much more complicated nonlinear differential equations and boundary value problems. However, if the problem becomes nonlinear, then the RKHSM does not require discretization or perturbation and it does not make closure approximation. Results of numerical examples show that the present method is an accurate and reliable analytical method for the seventh-order boundary value problem.

Appendix

Proof of Theorem 3. Let 𝑓∈𝑇⁸₂[0,1]. By Definition 2 we have 𝑓,𝑅_𝑦𝑇82=⁷_𝑖=0𝑓^(𝑖)(0)𝑅_𝑦^(𝑖)(0)+¹₀𝑓⁽⁸⁾(𝑥)𝑅_𝑦⁽⁸⁾(𝑥)𝑑𝑥. (A.1) Through several integrations by parts for (A.1) we have 𝑓,𝑅_𝑦𝑇82=⁷_𝑖=0𝑢^(𝑖)(0)𝑅_𝑦^(𝑖)(0)−(−1)^(7−𝑖)𝑅_𝑦^(15−𝑖)(0+_7𝑖=0)(−1)^(7−𝑖)𝑢^(𝑖)(1)𝑅_𝑦^(15−𝑖)(1)+¹₀𝑢(𝑥)𝑅_𝑦⁽¹⁶⁾(𝑥)𝑑𝑥. (A.2) Note that property of the reproducing kernel 𝑢,𝑅_𝑦𝑇82=𝑢𝑦. (A.3) Now, if 𝑅_𝑦⁽⁴⁾(0)+𝑅_𝑦⁽¹¹⁾(0)𝑅=0,_𝑦(5)(0)−𝑅_𝑦⁽¹⁰⁾(0)𝑅=0,_𝑦(6)(0)+𝑅_𝑦⁽⁹⁾(0)𝑅=0,_𝑦(7)(0)−𝑅_𝑦⁽⁸⁾(0)𝑅=0,_𝑦(8)(1)𝑅=0,_𝑦(9)(1)𝑅=0,_𝑦(10)(1)𝑅=0,_𝑦(11)(1)𝑅=0,_𝑦(12)(1)=0, (A.4) then (A.2) implies that 𝑅_𝑦⁽¹⁶⁾(𝑥)=𝛿𝑥−𝑦, (A.5) when 𝑥≠𝑦, then 𝑅_𝑦⁽¹⁶⁾(𝑥)=0, (A.6) and therefore 𝑅_𝑦(𝑥)=^16𝑖=1𝑐_𝑖𝑦𝑥^𝑖−1,𝑥≤𝑦,_16𝑖=1𝑑_𝑖𝑦𝑥^𝑖−1,𝑥>𝑦. (A.7) Since 𝑅_𝑦⁽¹⁶⁾(𝑥)=𝛿𝑥−𝑦, (A.8) we have 𝑅_𝑦^(𝑘)₊𝑦=𝑅_𝑦^(𝑘)₋𝑦𝑅,𝑘=0,…,14,_𝑦(15)+𝑦−𝑅_𝑦⁽¹⁵⁾₋𝑦=1. (A.9) Since 𝑅_𝑦∈𝑇⁸₂[0,1], it follows that 𝑅_𝑦(0)=𝑅^′_𝑦(0)=𝑅^′′_𝑦(0)=𝑅_𝑦⁽³⁾(0)=𝑅_𝑦(1)=𝑅^′_𝑦(1)=𝑅^′′_𝑦(1)=0. (A.10) From (A.4)–(A.10), the unknown coefficients 𝑐_𝑖(𝑦) and 𝑑_𝑖(𝑦)(𝑖=1,2,…,16) can be obtained. This completes the proof.

Disclosure

This paper is a part of the Ph.D. thesis of Ali Akgül.

Conflict of Interests

The authors declare that they do not have any competing or conflict of interests.