A Regularization-Free Scheme for Recovering Large External Forces of Higher-Order Nonlinear Evolution Equations

Abstract

In this study, the inverse engineering problems of the Ostrovsky equation (OE), Kawahara equation (KE), modified Kawahara equation (mKE), and sixth-order Korteweg-de Vries (KdV) equation will be investigated numerically. An effective numerical approach to tackle these inverse Coriolis dispersion problems and the above-mentioned inverse problems are still not available. To use different boundary shape functions, we must deal with the boundary data, initial conditions, and terminal time conditions of the OE, KE, mKE, and sixth-order KdV equations. The unknown Coriolis dispersion of OE and unknown large external forces of those three equations can be retrieved through back-substitution of the solution into the OE, KE, mKE, and sixth-order KdV equations while we obtain the solution with the symmetry property by employing the boundary shape function scheme (BSFS). Five numerical experiments with noisy data are carefully validated and discussed.

1. Introduction

Many researchers are interested in the direct and inverse problems of nonlinear evolution partial differential equations; however, only few researchers are concerned about the inverse problems because of their inherent ill-posed properties. Recently, nonlinear evolution equations [1] in mathematical physics play a major role in various fields, such as fluid mechanics, plasma physics, optical fibers, solid state physics, chemical kinematics, chemical physics, and geochemistry. In 1978, Ostrovsky derived an equation for weakly nonlinear surface and internal waves in a rotating ocean [2,3,4,5]. Many direct methods for obtaining the Ostrovsky equation (OE) have been resolved, such as the auxiliary equation method [6], the Exp-function method [7], the Fourier restriction norm method [8], the energy equation method [9], the improved tanh function method [10,11], the linear stability analysis [12], the mapping method [13], and the data-to-solution map [14].

The Cauchy problem for the quadratic and cubic OE with negative dispersion was studied in [15]. By using the Strichartz estimates instead of the Cauchy–Schwarz inequalities, they gave an alternative proof and only concentrated on the theoretical analysis. Yan et al. [16] solved the Cauchy problem for the cubic OE. They proved that the problem was locally well-posed for the initial data. Review [17] presented the theoretical, numerical, and experimental results of a study of the structure and dynamics of weak nonlinear internal waves in a rotating ocean accumulated over the past 40 years since the time when the approximate equation, called the OE, was derived in 1978. The relationship of this equation with other well-known wave equations, the integrability of the OE, and the condition for the existence of stationary solitary waves and envelope solitary waves were discussed. Coclite and di Ruvo [18] pondered an OE that included the regularized short pulse, the Korteweg-de Vries (KdV), and the modified KdV equations. They proved the well-posedness of the solutions for the Cauchy problem associated with these equations. Esfahani and Levandosky [19] proved that ground state solutions exist, which minimize the action among all nontrivial solutions, and that they used this variational characterization to study their stability. They also introduced a numerical method for computing ground states based on their variational properties. For inverse Coriolis dispersion problems of OEs, to the author’s best knowledge, there has been no report in the literature that the numerical approaches for these problems can deal with them.

For the engineering and mathematical problems of the Kawahara equation (KE), modified Kawahara equation (mKE), and sixth-order KdV equation, many direct approaches for acquiring the explicit travelling solitary wave solutions to the above-mentioned nonlinear evolution equations have been addressed, such as the algebraic method [20,21], the tanh method and the extended tanh method [22], the Adomian decomposition method [23], the Sine–Cosine method [24], the variational iteration method and the homotopy perturbation method [25,26], the dual-Petrov–Galerkin algorithm [27], the Crank–Nicolson differential quadrature algorithms [28], the variational iteration method and the Adomian decomposition method [29], the travelling wave ansatz [30], the multiplier techniques and compactness arguments [31], the general well-posedness principle [32], the optimal homotopy asymptotic method [33], the lattice Boltzmann model [34], the septic B-spline collocation method [35], the local radial basis functions method [36], the Crank-Nicolson discretization algorithm and the fifth-order quintic B-spline based differential quadrature method [37], the Kernel smoothing method and the Crank–Nicolson method [38], the complex method [39], and the generalized tanh–coth method [40]. The above-mentioned references did not mention the numerical solutions with noisy effect and did not demonstrate their real applications. For retrieving large external forces of KE, mKE, and sixth-order KdV equations, to the author’s best knowledge, there has been no published report in the literature that the numerical algorithms for these problems can tackle them. Our aim is to propose a method that can deal with the ill-posed problems without any regularization skills. In addition, we can demonstrate that this current scheme is adapted to tackle the inverse source issue of the higher-order nonlinear evolution equation by utilizing the boundary shape functions scheme (BSFS). We also use the techniques of superposition of the varied order boundary shape functions to cope with the inverse issue.

The rest of this study is organized as follows. Section 2 presents the nonlinear issue description and establishes the new boundary shape function, and boundary data of those problems. In Section 3, we attain shape functions and apply a free parameter into the boundary shape function, which results in different boundary shape functions. Five numerical experiments of the large external forces of higher-order nonlinear evolution equations are displayed in Section 4. Finally, we demonstrate the conclusions in Section 5.

2. Problem Description and Construction of Boundary Shape Function

For the unknown large external force of higher-order nonlinear evolution equation, we ponder the inverse source problem to manifest an unknown force function $P\left(x,t\right)$. They are seriously ill-posed problems in the engineering fields.

$$\begin{gathered} \varphi v_t\left(x,t\right)+\alpha v_{xt}\left(x,t\right)+\beta v_{xxxxxx}\left(x,t\right)+\mathsf{\gamma }{v}_{xxxxx}\left(x,t\right)+\mathsf{\epsilon }{v}_{xxxx}\left(x,t\right)+\mathsf{\xi }{v}_{xxx}\left(x,t\right)+\tau v_{xx}\left(x,t\right) \\ +\lambda v^n\left(x,t\right)v_x\left(x,t\right)=P\left(x,t\right), \left(x,t\right)\in \Gamma , \end{gathered}$$

(1)

$$v\left(x,0\right)=e\left(x\right), 0<x<m,$$

(2)

$$v\left(0,t\right)=0, v_{xx}\left(0,t\right)=0, v\left(m,t\right)=0, v_{xx}\left(m,t\right)=0, t\in \left[0,t_f\right],$$

(3)

where $\Gamma :=\left\{\left(x,t\right)\right|0<x<m, 0<t\le t_f\}$, $\varphi$, $\alpha$, $\beta$ $\mathsf{\gamma }$, $\mathsf{\epsilon }$, $\mathsf{\xi }$, $\tau$, and $n$ are arbitrary constants; $x$ and $t$ display spatial and time variables; $t_f$ is the final time, $v^n{v}_x$ is the nonlinear disturbance term; and $v_{xx}$, $v_{xxx}$, $v_{xxxx}$, $v_{xxxxx}$ and $v_{xxxxxx}$ are the dispersion terms of order two, three, four, five, and six, respectively. We will investigate the following four nonlinear partial differential equations (PDEs). When $\varphi =\beta =\mathsf{\gamma }=\mathsf{\xi }=0$ and $\alpha =\mathsf{\epsilon }=\tau =\lambda =n=1$, Equation (1) is called the Ostrovsky equation (OE). When $\alpha =\beta =\mathsf{\epsilon }=\tau =0$, $\varphi =\mathsf{\xi }=\lambda =n=1$, and $\mathsf{\gamma }=-1$, Equation (1) is called the Kawahara equation (KE). When $\alpha =\beta =\mathsf{\epsilon }=\tau =0$, $\varphi =\mathsf{\xi }=\lambda =1$, $\mathsf{\gamma }=-1$, and $n=2$, Equation (1) is called the modified Kawahara equation (mKE). When $\alpha =\mathsf{\gamma }=\mathsf{\epsilon }=\mathsf{\xi }=\tau =\lambda =0$, $\varphi =1$, $\beta =-1$, Equation (1) is called the sixth-order KdV equation. Moreover, Equation (2) is the initial condition, and Equation (3) is the boundary condition. The force function $P\left(x,t\right)$ is obtained after the substitution of solution $v\left(x,t\right)$ into the governing equations.

As Equations (1)–(3) have an unknown force function $P\left(x,t\right)$, we cannot resolve $v\left(x,t\right)$ directly. In order to retrieve $P\left(x,t\right)$ and resolve $v\left(x,t\right)$ in the whole region, we interpret the following additional information of

$$v\left(x,t_f\right)=r\left(x\right),$$

(4)

$$v_x\left(m,t\right)=\mu \left(t\right),$$

(5)

In the beginning, we ponder a partial boundary shape function where

$$E^0\left(x,t\right)=\frac{t^2}{t_f^2}r\left(x\right)+\left(1-\frac{t^2}{t_f^2}\right)e\left(x\right),$$

(6)

which spontaneously gratifies Equations (2) and (4) as follows:

$$E^0\left(x,0\right)=e\left(x\right), E^0\left(x,t_f\right)=r\left(x\right).$$

(7)

Presuming

$$E^0\left(x,t\right)\ne 0,$$

(8)

which indicates that functions $e\left(x\right)$ and $r\left(x\right)$ cannot be zeros at the same time.

We defined the following four polynomial-type shape functions $f_i\left(x\right),i=1,\dots ,4$

$$f_1\left(0\right)=1, f_1^{″}\left(0\right)=0, f_1\left(m\right)=0, f_1^{″}\left(m\right)=0,$$

(9)

$$f_2\left(0\right)=0, f_2^{″}\left(0\right)=1, f_2\left(m\right)=0, f_2^{″}\left(m\right)=0,$$

(10)

$$f_3\left(0\right)=0, f_3^{″}\left(0\right)=0, f_3\left(m\right)=1, f_3^{″}\left(m\right)=0,$$

(11)

$$f_4\left(0\right)=0, f_4^{″}\left(0\right)=0, f_4\left(m\right)=0, f_4^{″}\left(m\right)=1.$$

(12)

For those four boundary conditions in Equation (3), we reveal that they can be spontaneously satisfied by a nonzero boundary shape function as follows:

$$E\left(x,t\right)=E^0\left(x,t\right)-f_1\left(x\right)E^0\left(0,t\right)-f_2\left(x\right)E_{xx}^0\left(0,t\right)-f_3\left(x\right)E^0\left(\mathcal{l},t\right)-f_4\left(x\right)E_{xx}^0\left(m,t\right).$$

(13)

Through some calculations, we can demonstrate the following results:

$$f_1\left(x\right)=1+{\left(\frac{x}{m}\right)}^4-2{\left(\frac{x}{m}\right)}^3,$$

(14)

$$f_2\left(x\right)=\frac{x^2}{2}+\frac{m^2}{3}{\left(\frac{x}{m}\right)}^4-\frac{5m^2}{6}{\left(\frac{x}{m}\right)}^3,$$

(15)

$$f_3\left(x\right)=-{\left(\frac{x}{m}\right)}^4+2{\left(\frac{x}{m}\right)}^3,$$

(16)

$$f_4\left(x\right)=-\frac{m^2}{6}{\left(\frac{x}{m}\right)}^3+\frac{m^2}{6}{\left(\frac{x}{m}\right)}^4.$$

(17)

3. Conjunction of Boundary Shape Functions

We defined that the boundary shape function $E\left(x,t\right)$ is totally resolved by the shape functions $f_i\left(x\right),i=1,\dots ,4$, initial conditions, and the terminal time datum of $v\left(x,t\right)$ in the region $\mathsf{\Omega }$. By demonstrating a new algorithm, we can extend $E\left(x,t\right)$ and utilize it to the higher-order nonlinear evolution Equation (1). For the inverse source problem of the higher-order nonlinear evolution equation, we employ the superposition of the varied order boundary shape functions to handle it. Recently, Liu and Chang [41] and Chang [42] have used similar skills to solve inverse problems of nonlinear external forces on vibrating Euler–Bernoulli beams.

Notice that we cannot employ $E\left(x,t\right)$ as the basis of the solution because no free parameter occurs in $E\left(x,t\right)$. We utilize the $p$th-order shape functions $f_i\left(p,x\right), i=1,\dots ,4$ to extend Equations (9)–(17), which are resolved by

$$f_1\left(p,0\right)=1, f_1^{″}\left(p,0\right)=0, f_1\left(p,m\right)=0, f_1^{″}\left(p,m\right)=0,$$

(18)

$$f_2\left(p,0\right)=0, f_2^{″}\left(p,0\right)=1, f_2\left(p,m\right)=0, f_2^{″}\left(p,m\right)=0,$$

(19)

$$f_3\left(p,0\right)=0, f_3^{″}\left(p,0\right)=0, f_3\left(p,m\right)=1, f_3^{″}\left(p,m\right)=0,$$

(20)

$$f_4\left(p,0\right)=0, f_4^{″}\left(p,0\right)=0, f_4\left(p,m\right)=0, f_4^{″}\left(p,m\right)=1.$$

(21)

We can obtain the following derivations via some calculations:

$$f_1\left(p,x\right)=1+\frac{p+1}{2}{ŵ}^{p+3}-\frac{p+3}{2}{ŵ}^{p+2},$$

(22)

$$f_2\left(p,x\right)=\frac{x^2}{2}+\frac{m^2}{4}\frac{p\left(p+3\right)}{p+2}{ŵ}^{p+3}-\frac{m^2}{4}\frac{\left(p+1\right)\left(p+4\right)}{p+2}{ŵ}^{p+2},$$

(23)

$$f_3\left(p,x\right)=-\frac{p+1}{2}{ŵ}^{p+3}+\frac{p+3}{2}{ŵ}^{p+2},$$

(24)

$$f_4\left(p,x\right)=-\frac{m^2}{2\left(p+2\right)}{ŵ}^{p+2}+\frac{m^2}{2\left(p+2\right)}{ŵ}^{p+3},$$

(25)

in which the normalized coordinate $ŵ$ is described as

$$ŵ=\frac{x}{m}.$$

(26)

We can extend $E\left(x,t\right)$ in Equation (13) to the boundary shape functions as follows:

$$E\left(q,x,t\right)=E^0\left(p,x,t\right)-f_1\left(p,x\right)E^0\left(p,0,t\right)-f_2\left(p,x\right)E_{xx}^0\left(p,0,t\right)$$

$$-f_3\left(p,x\right)E^0\left(p,m,t\right)-f_4\left(p,x\right)E_{xx}^0\left(p,m,t\right),$$

(27)

in which

$$E^0\left(p,x,t\right)=\left(1-{ě}^{p+1}\right)e\left(x\right)+{ě}^{p+1}r\left(x\right),$$

(28)

$$ě=\frac{t}{t_f}.$$

(29)

4. Numerical Method and Examples

Presuming

$$v\left(x,t\right)={\displaystyle \sum }_{p=1}^s{b}_{p}E\left(p,x,t\right),$$

(30)

where $b_p$ are unknown coefficients to be obtained, and $E\left(p,x,t\right), p=1,\dots ,s$ are nonzero bases [42].

The condition $\mu \left(t\right)=v_x\left(m,t\right)$ is displayed in Equation (5). Hence, by employing the differential of Equation (30) with respect to $x$ and presenting $x=m$, we can obtain

$${\displaystyle \sum }_{p=1}^s{b}_p{E}_x\left(p,m,t\right)=\mu \left(t\right).$$

(31)

By collocating points $t_i=i{t}_f/j, i=1,\dots ,j$ to fit Equation (31), the linear equations can be obtained as follows:

$${\displaystyle \sum }_{p=1}^s{b}_{p}D{E}_x\left(p,m,t_i\right)=\mu \left(t_i\right),i=1,\dots ,j,$$

(32)

$${\displaystyle \sum }_{p=1}^s{b}_p=1,$$

(33)

where the last equation is employed to assure that $v\left(x,t\right)$ in Equation (30) can gratify the data in Equations (2)–(4).

By dealing with the $j_m=j+1$ linear Equations (32) and (33), we can choose the $s$ weight coefficients $b_p,p=1,\dots ,s$. Moreover, we can retrieve $P\left(x,t\right)$ by back replacing $v\left(x,t\right)$ with Equation (1), as follows:

$$\begin{array}{cc}\hfill P\left(x,t\right)=& \varphi {\displaystyle \sum }_{p=1}^s{b}_p{E}_t\left(p,x,t\right)+\alpha {\displaystyle \sum }_{p=1}^s{b}_p{E}_{xt}\left(p,x,t\right)+\beta {\displaystyle \sum }_{p=1}^s{b}_p{E}_{xxxxxx}\left(p,x,t\right)+\hfill \\ \hfill & \gamma {\displaystyle \sum }_{p=1}^s{b}_p{E}_{xxxxx}\left(p,x,t\right)+\epsilon {\displaystyle \sum }_{p=1}^s{b}_p{E}_{xxxx}\left(p,x,t\right)+\mathsf{\xi }{\displaystyle \sum }_{p=1}^s{b}_p{E}_{xxx}\left(p,x,t\right)+\hfill \\ \hfill & \tau {\displaystyle \sum }_{p=1}^s{b}_p{E}_{xx}\left(p,x,t\right)+\lambda {\displaystyle \sum }_{p=1}^s{b}_p{E}^n{E}_x\left(p,x,t\right).\hfill \end{array}$$

(34)

We can manifest that this proposed algorithm is fit to address the inverse source problem of the higher-order nonlinear evolution equation by employing the BSFS. As a matter of fact, $s$ is a small number so we simply want to resolve an $s$-dimensional small-scale normal linear system to regain $P\left(x,t\right)$.

Equations (4) and (5) are usually infected by random disturbance such as

$$\widehat{r}\left(x\right):=r\left(x\right)+zY\left(x\right), \widehat{\mu }\left(t\right):=\mu \left(t\right)+zY\left(t\right),$$

(35)

where $Y\left(x\right), Y\left(t\right)\in \left[-1,1\right]$ are random functions and $z$ is the strength of noisy influence.

To add the disturbance with a small $z$ value might be nonsensical when the maximum values of $\left|r\left(x\right)\right|$ and $\left|\mu \left(t\right)\right|$ are large. Hence, instead of Equation (35), in several events, the noise with a relative strength $\rho$ can be contemplated by

$$\widehat{r}\left(x\right):=r\left(x\right)+\rho \left|r\left(x\right)\right|Y\left(x\right), \widehat{\mu }\left(t\right):=\mu \left(t\right)+\rho \left|\mu \left(t\right)\right|Y\left(t\right).$$

(36)

For solving the unknown large external forces of higher-order nonlinear evolution equations, we define the following root mean square error (RMSE_n):

$$RMS{E}_n=\sqrt{\frac{1}{N_1{N}_2}{\displaystyle \sum }_{i=1}^{N_1}{\displaystyle \sum }_{j=1}^{N_2}{\left[g^n\left(x_i,t_j\right)-g^e\left(x_i,t_j\right)\right]}^2.}$$

(37)

where n is the number of iterations, $g^e$ is the exact solution, and $g^n$ is the numerical solution.

The computed order of convergence (COC) is calculated by

$$\mathrm{COC}:=\mid \frac{\ln \left[RMS{E}_{n+1}/RMS{E}_n\right]}{\ln \left[RMS{E}_n/RMS{E}_{n-1}\right]}\mid$$

(38)

4.1. Example 1 of OE with First-Type Boundary Condition as Equation (3)

We often decide the absolute error in the following five numerical experiments $\left|P_j\left(x_p,t_p\right)-P\left(x_p,t_p\right)\right|$ to estimate the accuracy of a numerical solution, in which $P_j\left(x_p,t_p\right)$ and $P\left(x_p,t_p\right)$ are the numerically recovered force and the exact force at $N$ points $\left(x_p,t_p\right), p=1,\dots ,N$, respectively. We also define a relative root mean square error as

$$e\left(P\right)=\sqrt{\frac{{{\displaystyle \sum }}_{p=1}^N{\left[P_j\left(x_p,t_p\right)-P\left(x_p,t_p\right)\right]}^2}{{{\displaystyle \sum }}_{p=1}^M{P}^2\left(x_p,t_p\right)}}$$

to estimate the accuracy of numerically retrieved force. The flowchart of this current scheme is shown in Figure 1.

We utilize Equations (2)–(3) and deliberate the following OE:

$$v_{xt}\left(x,t\right)+v_{xxxx}\left(x,t\right)+v_{xx}\left(x,t\right)+v\left(x,t\right)v_x\left(x,t\right)=P\left(x,t\right), \left(x,t\right)\in \Gamma ,$$

(39)

where $v\left(x,t\right)=2e^t\left(9x-6x^3-12x^4+9x^5\right)$ is the exact solution.

We employ the BSFS with a noise of $z=0.2$, $s=2$, and $j=100$, showing the numerically recovered solution of $P\left(x,t\right)$ with the exact force $P\left(x,t\right)$ in Figure 2a,b. Moreover, we display the maximum errors (MEs) of $v\left(x,t\right)$ and the retrieval of $P\left(x,t\right)$ under the obstreperous influence in Figure 3. Note that significant results are obtained with the Mes over the plane where $\left[0,1\right]\times \left(0,1\right]$ is $1.08\times {10}^{-2}$ for $v\left(x,t\right)$ and $7.04$ for $P\left(x,t\right)$. Moreover, we acquire the maximum absolute value of $P\left(x,t\right)$ as 3327.18, and $e\left(P\right)=2.40\times {10}^{-3}$. COC = 0.9464 is computed, which is near to a linear convergence.

To ponder a large relative disturbance of $\rho =0.5$ in Equation (36), we utilize the BSFS with $s=2$ and $j=100$ to recover$P\left(x,t\right)$, as demostrated in Figure 2c. The accuracy lost one order compared with the above-mentioned results, in which the ME for $v\left(x,t\right)$ is $0.3055$, the ME for $P\left(x,t\right)$ is 56.52, and $e\left(P\right)=7.78\times {10}^{-3}$. For this case, the CPU time is 0.5 s.

4.2. Example 2 of OE with Second-Type Boundary Condition

We demonstrate that the OE with a second-type boundary condition possesses the following boundary data (40), shape Function (41), and boundary shape Function (42):

$$v\left(0,t\right)=0, v_x\left(0,t\right)=0,v\left(m,t\right)=0, v_{xx}\left(m,t\right)=0,$$

(40)

$$f_1\left(p,x\right)=1+\frac{p}{2}{\overline{x}}^{p+2}-\frac{p+2}{2}{\overline{x}}^{p+1},$$

$$f_2\left(q,x\right)=x+\frac{p}{2m}{\overline{x}}^{p+2}-\frac{p+2}{2m}{\overline{x}}^{p+1},$$

$$f_3\left(p,x\right)=-\frac{p}{2}{\overline{x}}^{p+2}+\frac{p+2}{2}{\overline{x}}^{p+1},$$

$$f_4\left((p,x)\right)=-\frac{m^2}{2(p+1)}{\overline{x}}^{p+1}+\frac{m^2}{2(p+1)}{\overline{x}}^{p+2}$$

(41)

$$E\left(p,x,t\right)=E^0\left(p,x,t\right)-f_1\left(p,x\right)E^0\left(p,0,t\right)-f_2\left(p,x\right)E_x^0\left(p,0,t\right)-f_3\left(p,x\right)E^0\left(p,m,t\right)-f_4\left(p,x\right)E_{\mathrm{xx}}^0\left(p,m,t\right)$$

(42)

where $E^0\left(p,x,t\right)$ is depicted using Equation (28).

We apply Equation (2), Equation (40) and refer to Chang [42] and contemplate a complex space–time-dependent force generated from

$$\begin{array}{c}v\left(x,t\right)=1250e^t\left(8x^3-14x^4+6x^5\right)+3\cos \left(\pi t\right)[-2\left({\pi }^2+2\pi \right)\left(x^3-x^2\right)+8\pi \left(x^2-x\right)+\hfill \\ 8\exp \left[\sin \left(\pi x\right)\right]-8].\hfill \end{array}$$

(43)

In this case, we use the BSFS with a noise of $z=0.2$, $s=5$, and $j=10$ to illustrate the numerically recovered solution of $P\left(x,t\right)$ with the exact force $P\left(x,t\right)$ in Figure 4a,b. In addition, we exhibit the MEs of $v\left(x,t\right)$ and the retrieval of $P\left(x,t\right)$ under the obstreperous influence in Figure 5. Significant results are obtained with the MEs over the plane where $\left[0,1\right]\times \left(0,1\right]$ is $15.85$ for $v\left(x,t\right)$ and $64,597.88$ for $P\left(x,t\right)$, respectively. Furthermore, we attain the maximum absolute value of $P\left(x,t\right)$ as 3,220,412.97, and $e\left(P\right)=2.61\times {10}^{-2}$. COC = 0.9595 is computed, which is near to a linear convergence.

To consider a large relative disturbance of $\rho =0.03$ in Equation (36), we apply the BSFS with $s=5$ and $j=10$ to recover$P\left(x,t\right)$, as displayed in Figure 4c. The accuracy was not significantly different and was compared with the above-mentioned results, in which the ME for $v\left(x,t\right)$ is $26.74$, the ME for $P\left(x,t\right)$ is 143,421.83, and $e\left(P\right)=3.67\times {10}^{-2}$. For this instance, the CPU time is 1.5 s.

4.3. Example 3 of KE with Third-Type Boundary Condition

Considering the following KE with a third-type boundary condition, Equation (2), and referring to Chang [42] we get

$$v_t\left(x,t\right)+v_{xxxx}\left(x,t\right)+v_{xx}\left(x,t\right)+v\left(x,t\right)v_x\left(x,t\right)=P\left(x,t\right), \left(x,t\right)\in \Gamma ,$$

(44)

$$v\left(0,t\right)=0, v_x\left(0,t\right)=0, v_{xx}\left(\mathcal{l},t\right)=0, v_{xxx}\left(\mathcal{l},t\right)=0,$$

(45)

where $v\left(x,t\right)=e^t\left(6x^6-18m{x}^5+15m^2{x}^4\right)+3\left[t^2\sin \left(mt\right)-m{t}^3\cos \left(mt\right)\right]x^2+6\sin \left(xt\right)$ $-6xt+t^3\cos \left(mt\right)x^3$ is the exact solution.

Moreover, the boundary data (45), shape Function (46), and boundary shape Function (47) are as follows:

$$f_1\left(p,x\right)=1+\frac{p{m}^2}{\left(p+2\right)\left(p+3\right)}{\overline{x}}^{p+3}+\frac{x^2}{2}-\frac{m^2}{p+2}{\overline{x}}^{p+2},$$

$$f_2\left(p,x\right)=x+\frac{p{m}^2}{\left(p+2\right)\left(p+3\right)}{\overline{x}}^{p+3}+\frac{x^2}{2}-\frac{m^2}{p+2}{\overline{x}}^{p+2},$$

$$f_3\left(p,x\right)=-\frac{p{m}^2}{\left(p+2\right)\left(p+3\right)}{\overline{x}}^{p+3}+\frac{m^2}{p+2}{\overline{x}}^{p+2},$$

$$f_4\left(p,x\right)=-\frac{m^3}{\left(p+1\right)\left(p+2\right)}{\overline{x}}^{p+2}+\frac{m^3}{\left(p+2\right)\left(p+3\right)}{\overline{x}}^{p+3},$$

(46)

$$E\left(p,x,t\right)=E^0\left(p,x,t\right)-f_1\left(p,x\right)E^0\left(p,0,t\right)-f_2\left(p,x\right)E_x^0\left(p,0,t\right)$$

$$-f_3\left(p,x\right)E_{xx}^0\left(p,m,t\right)-f_4\left(p,x\right)E_{xxx}^0\left(p,m,t\right),$$

(47)

where $E^0\left(p,x,t\right)$ is described in Equation (28).

In this example, we utilize the BSFS with a noise of $z=0.2$, $s=2$, and $j=20$, exhibiting the numerically recovered solution of $P\left(x,t\right)$ with the exact force $P\left(x,t\right)$ in Figure 6a,b. Moreover, we demonstrate the MEs of $v\left(x,t\right)$ and the retrieval of $P\left(x,t\right)$ under the obstreperous influence in Figure 7. Excellent results are obtained with the MEs over the plane $\left[0,1\right]\times \left(0,1\right]$, where $v\left(x,t\right)$ is $5.36\times {10}^{-2}$ and $P\left(x,t\right)$ is $88.26$, respectively. Then, we acquire the maximum absolute value of $P\left(x,t\right)$ as 5717.39, and $e\left(P\right)=1.91\times {10}^{-2}$. COC = 0.9493 is computed, which is near to a linear convergence.

To deliberate a large relative disturbance of $\rho =0.2$ in Equation (36), we utilize the BSFS with $s=2$ and $j=20$ to recover$P\left(x,t\right)$, as demonstrated in Figure 6c. The accuracy lost one order compared with the above-mentioned results, where the ME for $v\left(x,t\right)$ is $0.59$, the ME for $P\left(x,t\right)$ is 429.02, and $e\left(P\right)=4.15\times {10}^{-2}$. For this example, the CPU time is 0.48 s.

4.4. Example 4 of mKE with Third-Type Boundary Condition as Equation (45)

In this study, we ponder the following mKE, Equation (2), Equation (45), and refer to Chang [42] to get

$$v_t\left(x,t\right)-v_{xxxx}\left(x,t\right)+v_{xxx}\left(x,t\right)+v^2\left(x,t\right)v_x\left(x,t\right)=P\left(x,t\right), \left(x,t\right)\in \Gamma ,$$

(48)

where $v\left(x,t\right)=e^{2t}\left(25m^2{x}^4-30m{x}^5+10x^6\right)$ is the exact solution.

In order to apply the BSFS with a noise of $z=0.2$, $s=2$, and $j=40$ we must show the numerically recovered solution of $P\left(x,t\right)$ with the exact force $P\left(x,t\right)$ in Figure 8a,b. In addition, we demonstrate the MEs of $v\left(x,t\right)$ and the retrieval of $P\left(x,t\right)$ under the obstreperous influence in Figure 9. Significant results are attained with the MEs over the plane $\left[0,1\right]\times \left(0,1\right]$, where $v\left(x,t\right)$ is $0.21$ and $P\left(x,t\right)$ is $1872.15$, respectively. Moreover, we obtain the maximum absolute value of $P\left(x,t\right)$ as 74,330.49, and $e\left(P\right)=2.42\times {10}^{-2}$. COC = 0.9523 is computed, which is near to a linear convergence.

To ponder a large relative disturbance of $\rho =0.4$ in Equation (36), we show the BSFS with $s=2$ and $j=40$ to recover$P\left(x,t\right)$, as displayed in Figure 8c. The accuracy compared with the above-mentioned results is close, where the ME for $v\left(x,t\right)$ is $0.52$, the ME for $P\left(x,t\right)$ is 2896.35, and $e\left(P\right)=3.93\times {10}^{-2}$. For this case, the CPU time is 0.6 s.

4.5. Example 5 of Sixth-Order KdV Equation with First-Type Boundary Condition as Equation (3)

We contemplate the following sixth-order KdV equation, Equations (2) and (3), and refer to Chang [42] to get

$$v_t\left(x,t\right)-v_{xxxxxx}\left(x,t\right)=P\left(x,t\right), \left(x,t\right)\in \Gamma ,$$

(49)

where $v\left(x,t\right)=30e^{2t}\left[800\left(2x^6-3x^5+x\right)+\sin \left(2\pi x\right)\right]+6{\pi }^{2}t\left(3x^5-5x^4+2x\right)$ is the exact solution.

In this numerical experiment, we utilize the BSFS with $z=0.9$, $s=6$, and $j=100$, displaying the numerically recovered solution of $P\left(x,t\right)$ with the exact force $P\left(x,t\right)$ in Figure 10a,b. In addition, we exhibit the MEs of $v\left(x,t\right)$ and the retrieval of $P\left(x,t\right)$ under the obstreperous influence in Figure 11. Significant results are attained with the MEs over the plane $\left[0,1\right]\times \left(0,1\right]$, where $v\left(x,t\right)$ is $5.49\times {10}^{-2}$ and $P\left(x,t\right)$ is $21,971.12$, respectively. Furthermore, we attain the maximum absolute value of $P\left(x,t\right)$ as $2.48\times {10}^8$, and $e\left(P\right)=9.36\times {10}^{-5}$. COC = 0.985 is computed, which is near to a linear convergence.

To consider a large relative disturbance of $\rho =0.05$ in Equation (36), we employ the BSFS with $s=6$ and $j=100$ to recover$P\left(x,t\right)$, as shown in Figure 10c, in which the ME for $v\left(x,t\right)$ is $637.60$, the ME for $P\left(x,t\right)$ is 2,103,924.20, and $e\left(P\right)=7.13\times {10}^{-3}$. For this study, the CPU time is 1.4 s.

5. Discussion

In Example 1, we can find the rotational wave of OE occurs with the time increment until the final time, as shown in Figure 2. In Figure 3, we reveal that the robustness of the BSFS is significant so that we can obtain the accurate wave shape and unknown nonlinear external force under the noisy effect. In order to test a complicated space–time-dependent force by using the BSFS in Example 2, we also discover the rotational wave of OE occurs with the time increment until the terminal time, as shown in Figure 4. It is interesting to observe that the least maximum error of the wave of OE is at the final time in Figure 5 due to the decrease in wave motion. In Example 3, we can realize that the least wave motion of KE in the middle of the spatial location as displayed in Figure 6. In Figure 7, we reveal that the least maximum error of unknown nonlinear external force occurs at the beginning, and the maximum error increases with time. In Example 4, we can find that the largest unknown nonlinear external force of mKE occurs in the final spatial location, and that the least unknown nonlinear external force of mKE happens around the middle of the spatial location, as demonstrated in Figure 8. In Figure 9, we discover that the biggest maximum errors of the wave and the unknown nonlinear external force of mKE both occur at the final time because of the exponential function effect, and the unknown nonlinear external force increases dramatically at the final time. Finally, for Example 5, we can realize that the six-order dispersion term is so much larger than the v_t term that the unknown nonlinear external force is negative in Figure 10. It is interesting to show that the least maximum error of the unknown nonlinear external force is t = 0.8 in Figure 11. To sum up, as the proposed algorithm does not need to utilize any discretization methods, we can obtain the correct and complete wave shape and the unknown nonlinear external force without the discretization error.

6. Conclusions

By employing the boundary shape functions scheme, we coped with the difficult retrieved problems of discovering unknown forces burdened with the higher-order inverse nonlinear evolution PDEs with two kinds of conditions. To make a comparison with the other expensive methods (Landweber iteration method, iteratively regularized Gauss–Newton method, Levenberg–Marquardt algorithm, Newton’s iterative method, homotopy perturbation method, level set method, and so forth) and the other identification methods (Tikhonov regularization procedures, quasi-inversion methods, etc.), we do not require the a priori data, need to cope with the complicated iteration techniques, find the optimal regularization parameter, deal with the complex computational procedure, tackle stability problems, and initial guesses in order to acquire significant results. Based on those numerical experiments, we demonstrate that the current scheme is proper for the nonlinear external forces of higher-order inverse nonlinear evolution PDEs and significant computational efficiency, and even for raising the large random disturbance up to 50%. Moreover, to the author’s best knowledge, there has been no report in the literature that the numerical methods for those five problems can deal with them. The current algorithm can be expanded to deal with the higher-order multi-dimensional inverse nonlinear evolution PDEs and the higher-order multi-dimensional inverse nonlinear wave PDEs, and will be explained further in the near future.