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Solving nonlinear thirdorder threepoint boundary value problems by boundary shape functions methods
Advances in Difference Equations volumeÂ 2021, ArticleÂ number:Â 146 (2021)
Abstract
For nonlinear thirdorder threepoint boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified threepoint boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear thirdorder threepoint BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.
1 Introduction
Boundary value problems (BVPs) have a lot of applications, like engineering technique, control theory and optimization, the boundary layer of fluid mechanics, aeroelasticity, sandwich beam analysis and beam deflection theory, electromagnetic waves, theory of thin film and incompressible flows. While for the conditions for the existence and uniqueness of solutions of thirdorder BVPs one may refer to [1â€“3], for the existence and uniqueness of thirdorder threepoint BVPs one can read [4â€“6].
Some researchers studied and solved thirdorder BVPs with different boundary conditions, for instance, using finite difference method [7, 8], quintic splines [9], quartic splines [10], nonpolynomial spline [11], quartic Bsplines [12], Haar wavelets method [13], exponential trial functions method [14], method of fundamental solution [15, 16], and the backward substitution method [17]. With the advent of computers, it has gained importance to develop more accurate and efficient numerical methods to solve higherorder BVPs. There are plenty of the related studies in the secondorder and thirdorder BVPs that have appeared in the literature, see [18â€“21].
For the thirdorder threepoint nonlinear BVP, it is difficult to exactly satisfy the boundary conditions, unless one designs the algorithm to exactly satisfy all the boundary conditions. In the paper we propose new numerical methods for solving the nonlinear thirdorder threepoint BVPs, designing the algorithms to automatically satisfy the threepoint boundary conditions, which are based on a novel concept of boundary shape function.
We arrange the paper as follows. In Sect.Â 2, we introduce thirdorder threepoint nonlinear BVP with separated boundary conditions and general nonseparated boundary conditions. In Sect.Â 3, we construct a boundary shape function and prove that it can automatically satisfy the specified threepoint boundary conditions, where an arbitrary free function is existent in the boundary shape function. In Sect.Â 4, we develop the first numerical algorithm based on the collocation technique and the trial functions, which are generated from the boundary shape functions by asking the bases of solution to satisfy the threepoint boundary conditions automatically. The shooting method is introduced in Sect.Â 5, where two examples are given to compare the numerical results computed by the boundary shape function method (BSFM) in Sect.Â 4 to that obtained by the shooting method. Taking advantage of the new concept of boundary shape function, it is easy to develop the second iterative algorithm to solve the thirdorder nonlinear BVPs with threepoint boundary conditions in Sect.Â 6, where two numerical examples are tested. In Sect.Â 7, we extend the idea of boundary shape function to the nonlinear thirdorder BVP with general nonseparated threepoint boundary conditions. Finally, some conclusions are drawn in Sect.Â 8.
2 Thirdorder nonlinear boundary value problem
Let us consider a thirdorder nonlinear ordinary differential equation (ODE)
with separated threepoint boundary conditions
where \(0< a<\ell \) and \(b_{1}\), \(b_{2}\), and \(b_{3}\) are constants, which may be zeros.
The general nonseparated threepoint boundary conditions:
will be considered later, where \({\mathcal{L}}_{1}\), \({\mathcal{L}}_{2}\), and \({\mathcal{L}}_{3}\) are linear operators.
3 The boundary shape function
The boundary shape function (BSF) is such a function that it satisfies the prescribed boundary conditions automatically. Apparently, the exact solution of the considered BVP is one member of the BSF, since it must satisfy the boundary conditions. In general, the numerical method used to approximate the exact solution cannot guarantee to automatically satisfy the boundary conditions exactly. We are going to devise a powerful method to solve Eqs.Â (1)â€“(4) based on the boundary shape functions. The idea of boundary shape function was used to solve the BVP with multipoint boundary conditions [22] and the singularly perturbed BVP with Robin boundary conditions [23]. The present problem with thirdorder nonlinear ODE (1), which is endowed with threepoint boundary conditions (2)â€“(4) or the general nonseparated threepoint boundary conditions (5)â€“(7), is more difficult to be solved than the secondorder BVPs in [22, 23]. We pursue this issue by the newly constructed BSF, which is a novel method for solving Eqs.Â (1)â€“(4), not yet reported in the literature.
We seek the polynomial type shape functions \(s_{k}(x)\), \(k=1,2,3\), satisfying
Depending on the values of the coefficient matrix \(c_{ij}\), \(i,j=1,2,3\), there are many solutions of \(s_{k}(x)\), \(k=1,2,3\). However, we prefer the lower orders shape functions and do not go into the details to the derivations of \(s_{k}(x)\), \(k=1,2,3\). For each example, we will list \(s_{k}(x)\), \(k=1,2,3\), explicitly.
Theorem 1
For any free function \(f(x)\in {\mathcal{C}}^{2}[0,\ell ]\), if \(s_{k}(x)\), \(k=1,2,3\), satisfy Eqs.Â (8)â€“(10), then the boundary shape function \(B(x)\), given by
is existent and satisfies the separated boundary conditions:
Proof
The existence of \(B(x)\) is guaranteed by the existence of \(s_{k}(x)\), \(k=1,2,3\), and the existence of a free function \(f(x)\in {\mathcal{C}}^{2}[0,\ell ]\). We prove Eq.Â (12) at first. Inserting \(x=0\) into Eq.Â (11), we have
Taking the differential of Eq.Â (11) and then inserting \(x=0\) leads to
Similarly, taking twice differentials of Eq.Â (11) and then inserting \(x=0\) yields
From Eqs.Â (15)â€“(17) it follows that
With the aids of the first equations in Eqs.Â (8)â€“(10), Eq.Â (18) is reduced to
Thus, Eq.Â (12) was proven. The proofs of Eqs.Â (13) and (14) can be done similarly.â€ƒâ–¡
In Theorem 1, there exists a free function \(f(x)\), which will be very useful for the development of the novel algorithms to solve the nonlinear thirdorder threepoint BVPs. We have to stress that the shape functions \(s_{1}(x)\), \(s_{2}(x)\), \(s_{3}(x)\) are not unique; however, the lowerorder shape functions are preferred.
4 First numerical algorithm
Based on the concept of boundary shape functions, we can develop a new algorithm of the boundary shape function method (BSFM) to solve Eq.Â (1) under threepoint boundary conditions (2)â€“(4).
For Eq.Â (1), we first choose some suitable basis functions \(\phi _{j}(x), j\in {\mathbb{N}}\), which are linearly independent and complete. Then we have the following result.
Theorem 2
For any function \(\phi _{j}(x)\in {\mathcal{C}}^{3}[0,\ell ]\),
satisfies
Proof
It is a direct extension of Theorem 1 to the basis function \(f(x)=\phi _{j}(x)\) and the new basis \(B(x)=E_{j}(x)\).â€ƒâ–¡
Now we describe a simple method to solve \(y(x)\) in Eqs.Â (1)â€“(4). We assume that the solution \(y(x)\) can be expanded by a set of trial functions \(E_{j}(x)\):
where \(a_{j}\) are unknown coefficients to be determined, and
guarantees that \(y(x)\), expressed by Eq.Â (22), automatically satisfies threepoint boundary conditions (2)â€“(4). Due to this reason, we only need to consider Eq.Â (1) being satisfied by \(y(x)\) in Eq.Â (22).
Let \({\mathbf{a}}:=\{a_{j}\}\) whose dimension is m. The algorithm of the boundary shape function method (BSFM) for solving \(y(x)\) in Eqs.Â (1)â€“(4) is summarized as follows.

(i)
Derive \(s_{1}(x)\), \(s_{2}(x)\), \(s_{3}(x)\), give m, \(n_{c}\), the initial guess of \({\mathbf{a}}_{0}\) and the convergence criterion Ïµ, and then compute \(x_{i}=i\ell /(n_{c}+1)\), \(i=1,\ldots ,n_{c}\).

(ii)
For \(k=0,1,2,\ldots \)â€‰, we repeat the following steps: Solving the following algebraic equations system to obtain a:
$$\begin{aligned}& \sum_{j=1}^{m} a_{j} E_{j}'''(x_{i})=F \Biggl(x_{i},\sum_{j=1}^{m} a_{j} E_{j}(x_{i}),\sum _{j=1}^{m} a_{j} E_{j}'(x_{i}), \sum_{j=1}^{m} a_{j} E_{j}''(x_{i}) \Biggr), \quad i=1,\ldots ,n_{c}, \\& \sum_{j=1}^{m} a_{j}=1. \end{aligned}$$
Then we have \({\mathbf{a}}_{k+1}\). If \({\mathbf{a}}_{k+1}\) converges according to a given stopping criterion \(r_{k}:=\{\mathbf{a}}_{k+1}{\mathbf{a}}_{k}\< \epsilon \), then stop; otherwise, go to step (ii). When \({\mathbf{a}}:=\{a_{j}\}\) is found, \(y(x)\) is obtained from Eq.Â (22).
5 Shooting method
The shooting method is well known and well established to solve BVPs [24]. We briefly sketch the shooting method to solve Eqs.Â (1)â€“(4), which requires to adjust any two unknown initial values missed in Eq.Â (2) such that the resulting integrated values of \([y(a),y'(a),y''(a)]\) and \([y(\ell ),y'(\ell ),y'(\ell )]\) after integrating Eq.Â (1) from \(x=0\) to \(x=a\) and to \(x=\ell \) can match the boundary conditions at \(x=a\) in Eq.Â (3) and at \(x=\ell \) in Eq.Â (4).
For the purpose of demonstration, we assume that \(c_{12}\neq 0\) in Eq.Â (2) and let
denote two unknown initial values. Then, by Eq.Â (2), we can derive another unknown initial value
Then we apply the fictitious time integration method (FTIM) developed by Liu and Atluri [25] to find Î± and Î² to match other two boundary conditions in Eqs.Â (3) and (4), which is obtained by the following iterations:
where \(t_{k}=k\Delta t\) is a fictitious time and Î”t is a time increment, and \(R_{1}\) and \(R_{2}\) denote the residuals in terms of boundary conditions (3) and (4):
where \(R_{1}\) and \(R_{2}\) are implicit functions of Î± and Î².
The iterative algorithm, by using the shooting method to solve \(y(x)\) of Eqs.Â (1)â€“(4), is given as follows. (i) Give Î”t, Î½, the initial guesses of \(\alpha _{0}\) and \(\beta _{0}\) and the convergence criterion \(\varepsilon _{0}\), and \(\Delta x=\ell /N\). (ii) Repeat \(k=0,1,2,\ldots \)â€‰, integrating Eq.Â (1) by the fourthorder Rungeâ€“Kutta method (RK4) with \(y(0)=\alpha _{k}\), \(y'(0)=y'_{0}\), \(y''(0)=\beta _{k}\) and with \(N_{1}=\ell /a\) steps from \(x=0\) to \(x=a\) and with N steps from \(x=0\) to \(x=\ell \). Taking
if \(\alpha _{k+1}\) and \(\beta _{k+1}\) converge according to a given stopping criterion
then terminate the iteration; otherwise, go to (ii) for the next iteration.
Example 1
We first consider the following threepoint BVP [26]:
whose exact solution is
For this problem, we take the following trial functions:
Through some operations, we can obtain \(s_{1}(x)=x11x^{2}/4+3x^{3}/2\), \(s_{2}(x)=6x^{2}4x^{3}\) and \(s_{3}(x)=x^{3}/2x^{2}/4\). Under the following parameters \(a_{0}=2\), \(b_{0}=3\), \(m_{1}=5\) (\(m=11\)), \(n_{c}=100\), and \(\epsilon =10^{8}\) the algorithm converges after 153 steps as shown in Fig.Â 1(a). From Fig.Â 1(b) we can find that the numerical solution \(y(x)\) is very close to the exact one with the maximum error (ME) being \(2.23\times 10^{11}\). The comparison of the errors in absolute values between the BSFM and other methods shows that the accuracy is much better than that computed in [27, 28] with about three orders, and than that computed by Li and Wu [29] with about five orders. In TableÂ 1, we list the numerical solutions and the absolute errors at different positions, where 1.31E(âˆ’11) means that the absolute error is \(1.31\times 10^{11}\), etc.
If we adjust the parameters to \(a_{0}=5\), \(b_{0}=1\) as those employed by Pandey [8], the algorithm BSFM converges after 55 steps and the ME is \(2.6\times 10^{11}\), which is also much better than that computed in [8].
By using the shooting method presented above, we take \(a_{0}=2\), \(b_{0}=3\), \(\Delta t=0.01\), \(\nu =50\), \(N=100\), \(R_{1}=y(1/2)\), \(R_{2}=y'(1)\), and the initial guesses \(\alpha =0\) and \(\beta =0\), which is convergence with 43 iterations under \(\varepsilon _{0}=10^{10}\) and the maximum error is \(1.51\times 10^{2}\). It can be seen that the convergence speed of the shooting method is faster than that of the BSFM; however, the accuracy of the BSFM with \(\mathrm{ME}=2.23\times 10^{11}\) is much better than that of the shooting method with about nine orders as shown in Fig.Â 2(a).
Example 2
Then we consider the following threepoint BVP [28, 30]:
whose exact solution is
We apply the BSFM to solve this problem with the following trial functions:
and the solution is expressed by
where \(E_{j}(x)\) were given by Eq.Â (20).
We can obtain \(s_{1}(x)=13x+2x^{2}\), \(s_{2}(x)=4x4x^{2}\), and \(s_{3}(x)=2x^{2}x\). Under the following parameters \(m=4\), \(n_{c}=50\), and \(\epsilon =10^{10}\), the algorithm converges after two steps as shown in Fig.Â 3(a). From Fig.Â 3(b) we can find that the numerical solution \(y(x)\) is very close to the exact one with ME being \(7.69\times 10^{13}\). The accuracy is much better than that computed in [28, 30] with about six orders.
By using the shooting method, we take \(\Delta t=0.01\), \(\nu =300\), \(N=100\), \(R_{1}=y(1/2)\), \(R_{2}=y(1)\), and the initial guesses \(\alpha =0\) and \(\beta =0\), which is convergence with 80 iterations under \(\varepsilon _{0}=10^{10}\) and the maximum error is \(5.17\times 10^{5}\). It can be seen that both the convergence speed of the BSFM is faster than that of the shooting method and the accuracy of the BSFM with \(\mathrm{ME}=7.69\times 10^{13}\) is much better than that of the shooting method with about eight orders as shown in Fig.Â 2(b).
6 Second numerical algorithm
On the other hand, based on the concept of boundary shape function, we can develop another iterative algorithm to solve Eqs.Â (1)â€“(4). According to Theorem 1, \(B(x)\) given in Eq.Â (11) satisfies the threepoint boundary conditions in Eqs.Â (2)â€“(4). Thus, when we consider the variable transformation from \(y(x)\) to a new variable \(u(x)\),
for any function \(u(x)\in {\mathcal{C}}^{3}[0,\ell ]\), the above \(y(x)\) automatically satisfies boundary conditions (2)â€“(4).
Inserting Eq.Â (36) for \(y(x)\) into Eq.Â (1), we generate a new thirdorder ODE for \(u(x)\):
which can be viewed as an initial value problem (IVP), whose initial values are given arbitrarily, say, \(u(0)=u'(0)=u''(0)=0\). However, there are a number of unknown parameters \(u(a)\), \(u'(a)\), \(u''(a)\), \(u(\ell )\), \(u'(\ell )\), \(u''(\ell )\) to be determined, which are collected as \({\mathbf{z}}:=[u(a), u'(a), u''(a), u(\ell ), u'(\ell ), u''(\ell )]^{\text{T}}\).
Letting
from Eq.Â (37) it follows that
which are subjected to the given initial conditions \(u_{1}(0)\), \(u_{2}(0)\), and \(u_{3}(0)\). Unfortunately, \({\mathbf{z}}:=[u_{1}(a), u_{2}(a), u_{3}(a), u_{1}(\ell ), u_{2}(\ell ), u_{3}( \ell )]^{\text{T}}\) is an unknown vector. If z is available, we can apply the RK4 to integrate the ODEs in Eq.Â (39) and then \(y(x)\) is obtained from Eq.Â (36).
The iterative algorithm BSFM for solving \(y(x)\) of Eqs.Â (1)â€“(4) is summarized as follows.

(i)
Derive \(s_{1}(x)\), \(s_{2}(x)\), \(s_{3}(x)\), give \(u_{1}(0)\), \(u_{2}(0)\), \(u_{3}(0)\), an initial guess of \({\mathbf{z}}_{0}\), and the convergence criterion Ïµ, and then compute \(\Delta x=\ell /N\) with N given.

(ii)
For \(k=0,1,2,\ldots \)â€‰, we repeat the following iterations: Applying the RK4 to integrate the following ODEs with \(N_{1}=a/\Delta x\) steps to \(x=a\), and N steps to \(x=\ell \):
$$\begin{aligned}& u_{1}'(x)=u_{2}(x), \\& u_{2}'(x)=u_{3}(x), \\& u_{3}'(x)=H\bigl(x,u_{1}(x),u_{2}(x),u_{3}(x);{ \mathbf{z}}_{k}\bigr). \end{aligned}$$
Taking
if \({\mathbf{z}}_{k+1}\) converges according to a given stopping criterion \(r_{k}:=\{\mathbf{z}}_{k+1}{\mathbf{z}}_{k}\< \epsilon \), then stop; otherwise, go to step (ii). When \(u(x)=u_{1}(x)\) is solved, \(y(x)\) is obtained from Eq.Â (36).
Example 3
We recast ExampleÂ 2 to a nonlinear one:
whose exact solution is given by Eq.Â (33).
We apply the above iterative algorithm BSFM to solve this problem with \(s_{1}(x)=13x+2x^{2}\), \(s_{2}(x)=4x4x^{2}\), and \(s_{3}(x)=2x^{2}x\). Under the following parameters \(u_{1}(0)=u_{2}(0)=u_{3}(0)=0\), \({\mathbf{z}}_{0}=(0,0)^{\text{T}}\), \(N=500\), and \(\epsilon =10^{10}\), the algorithm converges after 1148 iterations as shown in Fig.Â 4(a). From Fig.Â 4(b) we can find that the numerical solution \(y(x)\) is very close to the exact one with ME being \(7.42\times 10^{11}\). Although the nonlinear threepoint BVP is considered here, the accuracy is much better than that computed in [28, 30] with about four orders. In TableÂ 2, we list the numerical solutions and the absolute errors at different positions.
By using the shooting method, we take \(\Delta t=0.01\), \(\nu =200\), \(N=100\), \(R_{1}=y(1/2)\), \(R_{2}=y(1)\), and the initial guesses \(\alpha =0\) and \(\beta =0\), which is convergence with 71 iterations under \(\varepsilon _{0}=10^{10}\) and the maximum error is \(1.8\times 10^{2}\). It can be seen that both the convergence speed of the BSFM is faster than that of the shooting method and the accuracy of the BSFM with \(\mathrm{ME}=7.42\times 10^{11}\) is much better than that of the shooting method with about nine orders as shown in Fig.Â 5(a).
Example 4
We consider the following nonlinear threepoint BVP:
whose exact solution is given by \(y(x)=e^{x}\sin (\pi x)\) such that \(h(x)\) can be computed by inserting \(y(x)=e^{x}\sin (\pi x)\) into Eq.Â (41).
We apply the BSFM to solve this problem with \(s_{1}(x)=1x/2x^{2}/2\), \(s_{2}(x)=x/2x^{2}/2\) and \(s_{3}(x)=x/2+x^{2}/2\). Under the following parameters \(u_{1}(0)=u_{2}(0)=u_{3}(0)=0\), \({\mathbf{z}}_{0}=(0,0,0)^{\text{T}}\), \(N=200\), and \(\epsilon =10^{10}\) the algorithm converges after 31 iterations as shown in Fig.Â 6(a). From Fig.Â 6(b) we can find that the numerical solution \(y(x)\) is very close to the exact one with ME being \(6.24\times 10^{10}\). Although the nonlinear threepoint BVP is considered here, the accuracy is very good. In TableÂ 3, we list the numerical solutions and the absolute errors at different positions.
By using the shooting method, we take \(\Delta t=0.01\), \(\nu =90\), \(N=100\), \(R_{1}=y(1/2)y''(1/2)\pi ^{2} e^{1/2}\), \(R_{2}=y(1)\), and the initial guesses \(\alpha =0\) and \(\beta =0\), which is convergence with 85 iterations under \(\varepsilon _{0}=10^{10}\) and the maximum error is \(1.72\times 10^{3}\). It can be seen that both the convergence speed of the BSFM is faster than that of the shooting method and the accuracy of the BSFM with \(\mathrm{ME}=6.24\times 10^{10}\) is much better than that of the shooting method with about seven orders as shown in Fig.Â 5(b).
7 Nonseparated threepoint BVP
In this section we extend Theorem 1 to the nonseparated threepoint boundary conditions. The following example is used to demonstrate the new idea.
Example 5
We first consider the following nonseparated threepoint BVP [28] as a demonstrative case to introduce the concept of boundary shape function for the nonseparated threepoint boundary conditions:
whose exact solution is
We can apply the algorithm BSFM in Sect.Â 4 to solve this problem, where we can find
Now, letting
we can prove the following result.
Lemma 1
For any free function \(\phi _{j}(x)\in {\mathcal{C}}^{1}[0,1]\), if \(s_{k}(x)\), \(k=1,2,3\), satisfy
then the function \(E_{j}(x)\), given by Eq.Â (46), satisfies
Proof
The lowerorder solutions of \(s_{k}(x)\), \(k=1,2,3\), for Eq.Â (47) are given in Eq.Â (45). Inserting \(x=0\) into Eq.Â (46), we have
which, due to Eq.Â (47), reduces to
Taking the differential of Eq.Â (46) and inserting \(x=0\), we have
which, due to Eq.Â (47), reduces to
Taking the differential of Eq.Â (46) and inserting \(x=1/2\) and \(x=1\), we have
Subtracting the first equation by the second equation leads to
which, due to Eq.Â (47), reduces to
Thus, we end the proof of Eq.Â (48).â€ƒâ–¡
Now we apply the BSFM in Sect.Â 4 to solve ExampleÂ 5 with the following trial functions:
and the solution is expressed by
where \(E_{j}(x)\) were given by Eq.Â (46). The above \(y(x)\) automatically satisfies the boundary conditions in Eq.Â (43), due to Eq.Â (48) and \(\sum_{j=1}^{m} a_{j}=1\).
Under the following parameters \(m=4\), \(n_{c}=100\), and \(\epsilon =10^{10}\), the algorithm converges after eight steps as shown in Fig.Â 7(a). From Fig.Â 7(b) we can find that the numerical solution \(y(x)\) is very close to the exact one with ME being \(1.67\times 10^{11}\). The accuracy is much better than that computed in [28] with about four orders. In TableÂ 4, we list the numerical solutions and the absolute errors at different positions.
Inspired by LemmaÂ 1, we can prove the following result for Eq.Â (1), which is subjected to the nonseparated threepoint boundary conditions (5)â€“(7).
Theorem 3
For any free function \(f(x)\in {\mathcal{C}}^{2}[0,\ell ]\), if \(s_{k}(x)\), \(k=1,2,3\), satisfy
then the boundary shape function \(B(x)\), given by
satisfies the nonseparated threepoint boundary conditions:
Proof
Applying the linear operator \({\mathcal{L}}_{1}\) to Eq.Â (54) and using the linear property, we have
which with the help from Eqs.Â (51)â€“(53) becomes
This ends the proof of Eq.Â (55). The proofs of Eqs.Â (56) and (57) can be done similarly.â€ƒâ–¡
The above theorem is crucial for treating very complex nonseparated threepoint boundary conditions of thirdorder nonlinear BVP and for guaranteeing that the boundary conditions are satisfied exactly. Now we apply the iterative algorithm BSFM in Sect.Â 6 to the following problem, which has transformed the BVP to the corresponding IVP.
Example 6
We consider the following quadratic nonlinear nonseparated threepoint BVP:
whose exact solution is given by \(y(x)=e^{x}\sin (\pi x)\) such that \(h(x)\) can be computed by inserting \(y(x)=e^{x}\sin (\pi x)\) into Eq.Â (58).
We apply the iterative algorithm BSFM in Sect.Â 6 to solve this problem with \(s_{1}(x)=4+8x/34x^{2}/3\), \(s_{2}(x)=x/3x^{2}/2\), and \(s_{3}(x)=1x/3+2x^{2}/3\). For this problem, there are four unknown parameters \({\mathbf{z}}=[u(1/2),u'(1/2),u''(1/2),u(1)]^{\text{T}}\).
Under the following parameters \(u_{1}(0)=u_{2}(0)=u_{3}(0)=0\), \({\mathbf{z}}_{0}=(0,0,0,0)^{\text{T}}\), \(N=100\), and \(\epsilon =10^{10}\) the algorithm converges after 56 iterations as shown in Fig.Â 8(a). From Fig.Â 8(b) we can find that the numerical solution \(y(x)\) is very close to the exact one with ME being \(1.4\times 10^{8}\). Although the nonseparated threepoint nonlinear BVP is difficult to be treated by the numerical method, the accuracy of the problem we considered is very good. In TableÂ 5, we list the numerical solutions and the absolute errors at different positions.
Example 7
We consider the following cubic nonlinear nonseparated threepoint BVP:
whose exact solution is given by \(y(x)=e^{x}(xx^{2})\) such that \(h(x)\) can be computed by inserting \(y(x)=e^{x}(xx^{2})\) into Eq.Â (59).
We apply the BSFM to solve this problem with \(s_{1}(x)=9/344x/17+4x^{2}/17\), \(s_{2}(x)=4/17+4x/174x^{2}/17\), and \(s_{3}(x)=11/34+23x/343x^{2}/17\). For this problem there are four unknown parameters \({\mathbf{z}}=[u(1/2),u'(1/2),u(1),u'(1)]^{\text{T}}\).
Under the following parameters \(u_{1}(0)=u_{2}(0)=u_{3}(0)=0\), \({\mathbf{z}}_{0}=(0,0,0,0)^{\text{T}}\), \(N=100\), and \(\epsilon =10^{10}\), the algorithm converges after 21 iterations as shown in Fig.Â 9(a). From Fig.Â 9(b) we can find that the numerical solution \(y(x)\) is very close to the exact one with ME being \(8.91\times 10^{9}\). Although the nonseparated threepoint nonlinear BVP is difficult to be treated by the numerical method, the accuracy of the problem we considered is very good. In TableÂ 6, we list the numerical solutions and the absolute errors at different positions.
For Examples 6 and 7, we find that although the shooting method in Sect.Â 5 is modified, it cannot be applied to solve them, because there exist three unknown initial values \(y(0)\), \(y'(0)\), and \(y''(0)\) to be determined. In contrast, the presented BSFM can provide very accurate solutions of Examples 6 and 7. Although in the BSFM we also transformed the BVP to the corresponding IVP, the major difference is that the initial values are given with known values, say \(u(0)=u'(0)=u''(0)=0\), and we adjust the unknown terminal values at \(x=a\) and \(x=\ell \) through the integrations of the new ODEs. The process in the BSFM is much more stable than that in the shooting method with unknown initial values \(y(0)\), \(y'(0)\), and \(y''(0)\).
8 Conclusions
The paper is witnessed to derive the boundary shape functions to exactly satisfy the given separated threepoint boundary conditions and general nonseparated threepoint boundary conditions of the thirdorder nonlinear BVP. According to the new idea of boundary shape functions, we have developed two novel numerical algorithms used in the solutions of the thirdorder threepoint nonlinear BVPs. The main contributions are the introduction of a new concept of boundary shape functions and then deriving the new bases and a new variable transformation, which automatically satisfy all the boundary conditions specified. In the first algorithm, since the bases have already satisfied the threepoint boundary conditions, we only need to collocate points inside the domain to satisfy the governing equation, which is easily performed by solving the resulting algebraic equations system to determine the expansion coefficients. On the other hand, in the second algorithm we have transformed the nonlinear thirdorder threepoint BVP to the initial value problem of a new nonlinear ODE, which is convergent very fast, to find solution. Numerical examples confirmed and showed that the two novel algorithms based on the boundary shape functions are highly accurate and stable.
Availability of data and materials
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The authors would like to thank the referees and the editor for their valuable comments which led to an improvement of this work.
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This work is supported by the Fundamental Research Funds for the Central Universities (No. B200202126), the Natural Science Foundation of Jiangsu Province (No. BK20190073), the State Key Laboratory of Acoustics, Chinese Academy of Sciences (No. SKLA202001), and the China Postdoctoral Science Foundation (Nos. 2017M611669, 2018T110430).
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Lin, J., Zhang, Y. & Liu, CS. Solving nonlinear thirdorder threepoint boundary value problems by boundary shape functions methods. Adv Differ Equ 2021, 146 (2021). https://doi.org/10.1186/s1366202103288x
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DOI: https://doi.org/10.1186/s1366202103288x
Keywords
 Thirdorder nonlinear boundary value problems
 Threepoint boundary conditions
 Boundary shape functions methods