Solution of nonlinear problems by a new analytical technique using Daftardar-Gejji and Jafari polynomials

Abstract

This article shows the solution of nonlinear differential equations by a new analytical technique called modified optimal homotopy perturbation method. Daftardar-Gejji and Jafari polynomials are used in the proposed method for the expansion of nonlinear term in the equation. Four nonlinear boundary value problems of fourth, fifth, sixth, and eighth orders are solved by modified optimal homotopy perturbation method as well as optimal homotopy perturbation method. The achieved consequences are authenticated by comparison with the results gained by the existing method—optimal homotopy perturbation method. The method consists of few steps and gives better results. The easy applicability and fast convergence are goals of the applied technique. The applied technique has fewer limitations and can be used for the phenomena containing ordinary differential equation, partial differential equation, integro-differential equation, and their systems.

Introduction

Various physical phenomena can be formulated in terms of differential equations. These equations are used in different fields of science and engineering. Differential equations of different orders have their own importance and applications. Here, fourth-, fifth-, sixth-, and eighth-order differential equations are taken forming different phenomena.^1,2

Particular attention has been given to solve nonlinear differential equations of different forms. Many analytical and numerical techniques are applied to solve such problems, but still some improvement is needed.

Homotopy asymptotic method (HAM) was first introduced by S Liao,^3,4 keeping the homotopy concept in topology. Similarly, Ji-Huan He^5,6,7,8 used homotopy perturbation method (HPM)^9,10 by taking the idea of perturbation process and homotopy concept. These techniques are applied by many other researchers also, but need to be improved. V Marinca et al.^11,12,13,14 used a combined idea from the above-mentioned methods and created a new method called optimal homotopy asymptotic method (OHAM).^15,16 Similarly, the same authors used the concept of He’s homotopy perturbation technique and produced optimal homotopy perturbation method (OHPM) mentioned in Marinca and Herisanu,^17,18 Jajarmi et al., ¹⁹ and Alolhasani et al. ²⁰ Both of the above methods consist of homotopy, auxiliary functions, and optimal convergence control parameters. The nonlinear term in the equation is expanded in a series, and the parameters are found by the use of least square method or Galerekin’s method, but further struggles are needed to improve the results. Different researchers used variational iteration techniques to solve boundary value problems.^21–25 The new and most important task can be seen in previous studies.^26–33

The aim is achieved after many struggles, and it is found that the results can be improved by introducing new polynomials called Daftardar-Gejji and Jafari polynomials^26,27 in OHPM. These polynomials have a great role in rapid convergence and improving the results. Here, the nonlinear term in the equation is expanded in terms of Daftardar-Gejji and Jafari polynomials. The auxiliary parameters found in the auxiliary functions can be calculated using different methods. Here, least square and Galerekin’s methods are more suitable. The applied method has no limitations and can be used for the solution of ordinary differential equations (ODEs), partial differential equations (PDEs), integro-differential equations (IDEs), and their systems. Fourth-, fifth-, sixth-, and eighth-order nonlinear differential equations are solved by standard OHPM and modified optimal homotopy perturbation method (MOHPM) to authenticate the code. The achieved outcomes are also compared with the established outcomes of OHAM to check the potential of the used technique.

Sections “Introduction” and “Explanation of the polynomials used in OHPM and MOHPM” consist of introduction and explanation of the polynomials used in OHPM and MOHPM. The proposed method has been explained in section “Proposed method and materials.” Section “Application of MOHPM” is devoted to the application of OHPM and MOHPM to nonlinear differential equations. Finally, in section “Conclusion,” some conclusions are given.

Explanation of the polynomials used in OHPM and MOHPM

Explanation about the polynomials of OHPM and MOHPM

Suppose we have a nonlinear term $N(\mu )={\mu }^2$ in a functional equation such that

$$\mu ={\mu }_0+\ell {\mu }_1+{\ell }^2{\mu }_2+{\ell }^3{\mu }_3$$

If this function is expanded in Taylor series about ${\mu }_0$ and considered $\ell =1$ , then we have

$$\begin{array}{cc}\multicolumn{1}{c}{N\left(\mu \right)}& ={\mu }^2={({\mu }_0+{\mu }_1+{\mu }_2+{\mu }_3)}^2\hfill \\ \multicolumn{1}{c}{}& =\underbrace{{\mu }_0^2}+\underbrace{({\mu }_1^2+2{\mu }_0{\mu }_1)}\hfill \\ \multicolumn{1}{c}{}& +\underbrace{{({\mu }_2^2+2{\mu }_0{\mu }_2+2{\mu }_1\mu )}_2}\hfill \\ \multicolumn{1}{c}{}& +\underbrace{(2{\mu }_0{\mu }_3+2{\mu }_1{\mu }_3+2{\mu }_2{\mu }_3+{\mu }_3^2)}\hfill \end{array}$$

(1)

Relating to the power of $\ell$ , the OHPM built-in polynomials for this nonlinear term are

$$\begin{array}{c}{N}_0\left({\mu }_0\right)=\underbrace{{\mu }_0^2}\hfill \\ \multicolumn{1}{c}{N_1({\mu }_0,{\mu }_1)=\underbrace{2{\mu }_0{\mu }_1}}\\ \multicolumn{1}{c}{N_2({\mu }_0,{\mu }_1,{\mu }_2)=\underbrace{{\mu }_1^2+2{\mu }_0{\mu }_2}}\\ \multicolumn{1}{c}{N_3({\mu }_0,{\mu }_1,{\mu }_2,{\mu }_3)=\underbrace{2{\mu }_1{\mu }_2+2{\mu }_0{\mu }_3}}\end{array}$$

(2)

We notice that these polynomials took only six terms of the series while the series equation (1) consists 10 terms. Relating to the power of $\ell$ , the polynomials in MOHPM for this nonlinear term are

$$\begin{array}{c}{N}_0\left({\mu }_0\right)=\underbrace{{\mu }_0^2}\hfill \\ \multicolumn{1}{c}{N_1({\mu }_0,{\mu }_1)=\underbrace{{\mu }_1^2+2{\mu }_0{\mu }_1}}\\ \multicolumn{1}{c}{N_2({\mu }_0,{\mu }_1,{\mu }_2)=\underbrace{{\mu }_2^2+2{\mu }_0{\mu }_2+2{\mu }_1{\mu }_2}}\\ \multicolumn{1}{c}{N_3({\mu }_0,{\mu }_1,{\mu }_2,{\mu }_3)=\underbrace{2{\mu }_0{\mu }_3+2{\mu }_1{\mu }_3+2{\mu }_2{\mu }_3+{\mu }_3^2}}\end{array}$$

(3)

From the above, we notice that these polynomials took all the terms of the expansion equation (1). It clearly indicates the superiority of MOHPM polynomials (Daftardar-Gejji and Jafari polynomials) over the OHPM polynomials.

Convergence

Here the convergence of Daftardar-Gejji and Jafari polynomials is determined. The nonlinear function $N(\mu )$ is written in the following form

$$\begin{array}{cc}\multicolumn{1}{c}{N\left(\mu \right)=}& N\left({\mu }_0\right)+[N({\mu }_0+{\mu }_1)-N\left({\mu }_0\right)]\hfill \\ \multicolumn{1}{c}{}& +[N({\mu }_0+{\mu }_1+{\mu }_2)-N({\mu }_0+{\mu }_1)]+\cdots \hfill \end{array}$$

(4)

Let $G_0=N({\mu }_0)$ and

$$G_n=N\left(\sum _{i=0}^n{\mu }_i\right)-N\left(\sum _{i=0}^{n-1}{\mu }_i\right)$$

(5)

for $n=1,2,\dots$ . Therefore, $N(\mu )=\sum _{i=0}^{\infty }{G}_i$ .

Also, using Taylor’s theorem given below:

Taylor’s theorem: suppose that $\mu \in C^n(u),$ where $u$ is an open subset of $X$ (Banach space) containing the line segment from $x_0$ to $h$ , then

$$\begin{array}{c}\multicolumn{1}{c}{N(x_0+h)=N\left(x_0\right)+N\prime \left(x_0\right)h+N^{″}\left(x_0\right)\frac{\left(h^2\right)}{2!}}\\ +\cdots +N^{(n-1)}\left(x_0\right)\frac{\left(h^{(n-1)}\right)}{(n-1)!}\hfill \\ \multicolumn{1}{c}{=\sum _{k=0}^n{N}^{(k)}\left(x_0\right)\frac{h^k}{k!}+q\left(x\right)}\end{array}$$

(6)

where $q(x)$ is such that $\parallel q(x)\parallel =O\parallel x{\parallel }^n$ and $N^{(k)}(x)$ is symmetric.

we get as

$$\begin{array}{c}\multicolumn{1}{c}{G_1=N({\mu }_0+{\mu }_1)-N\left({\mu }_0\right)=N\left({\mu }_0\right)}\\ +N\prime \left({\mu }_0\right){\mu }_1+N^{″}\left({\mu }_0\right)\frac{{\mu }_1^2}{2!}+\cdots -N\left({\mu }_0\right)\hfill \\ \multicolumn{1}{c}{=\sum _{k=1}^{\infty }{N}^{(k)}\left({\mu }_0\right)\frac{{\mu }_1^k}{k!}}\end{array}$$

(7)

$$\begin{array}{cc}\multicolumn{1}{c}{G_2}& =N({\mu }_0+{\mu }_1+{\mu }_2)-N({\mu }_0+{\mu }_1)\hfill \\ \multicolumn{1}{c}{}& =N\left({\mu }_0\right)+N\prime \left({\mu }_0\right){\mu }_1+N^{″}\left({\mu }_0\right)\frac{{\mu }_1^2}{2!}+\cdots -N\left({\mu }_0\right)\hfill \\ \multicolumn{1}{c}{}& =\sum _{j=1}^{\infty }{N}^{(j)}({\mu }_0+{\mu }_1)\frac{{\mu }_2^j}{j!}\hfill \\ \multicolumn{1}{c}{}& =\sum _{j=1}^{\infty }\left[\sum _{i=0}^{\infty }{N}^{(i+j)}\left({\mu }_0\right)\frac{{\mu }_1^i}{i!}\right]\frac{{\mu }_2^i}{i!}\hfill \end{array}$$

(8)

$$G_3=\sum _{i_{3=1}}^{\infty }\sum _{i_{2=0}}^{\infty }\sum _{i_{1=0}}^{\infty }{N}^{(i_{1+}{i}_{2+}{i}_3)}\left({\mu }_0\right)\frac{{\mu }_3^{i_3}}{i_3!}\frac{{\mu }_2^{i_2}}{i_2!}\frac{{\mu }_1^{i_1}}{i_1!}$$

(9)

In general

$$G_n=\sum _{i_{n=1}}^{\infty }\sum _{i_{n-1=0}}^{\infty }\dots \sum _{i_{1=0}}^{\infty }\left[N^{\left(\sum _{k=1}^n{i}_k\right)}\left({\mu }_0\right)\underset{j=1}{\overset{n}{\Pi }}\frac{{\mu }_j^{i_j}}{i_j!}\right]$$

(10)

Theorem 1

If $N$ is $C^{\infty }$ in a neighborhood of ${\mu }_0$ and

$$\begin{array}{c}\parallel N^n\left({\mu }_0\right)\parallel =\mathrm{Sup}{N}^n\left({\mu }_0\right)({\hslash }_1,\dots {\hslash }_n):\parallel \hslash {\parallel }_i\le 1,\hfill \\ 1\le i\le n\le \Re \hfill \end{array}$$

(11)

For any $n$ and some real $\Re >0$ and $\parallel {\mu }_i\parallel \le M<1/e$ , the series $\sum _{n=0}^{\infty }{G}_n$ is absolutely convergent, $i=1,2,3,\dots$ . Moreover

$$\parallel G_n\parallel \le L{M}^n{e}^{n-1}(e-1),\mathrm{for}n=1,2,\dots$$

(12)

Proof

Consider equation (10)

$$\begin{array}{c}\parallel G_n\parallel \le \Re M^n\sum _{i_{n=1}}^{\infty }\sum _{i_{n-1=0}}^{\infty }\dots \sum _{i_{1=0}}^{\infty }\hfill \\ \left[N^{\left(\sum _{k=1}^n{i}_k\right)}\left({\mu }_0\right)\left(\underset{i_1=1}{\overset{n}{\Pi }}\frac{1}{i_j!}\right)\right]=\Re M^n{e}^{n-1}(e-1),n=1,2,3,\dots \hfill \end{array}$$

(13)

Thus, by the convergent series, $\parallel \sum _{n=1}^{\infty }{G}_n\parallel$ is dominated, $\Re M(e-1)\sum _{n=1}^{\infty }(\mathrm{Me}{)}^{n-1}$ , where $M\prec 1/e$ . Hence, by the comparison test, $\sum _{n=1}^{\infty }\parallel G_n\parallel$ is absolutely convergent. Now to present the boundedness of all $\parallel {\mu }_i\parallel$ , for all $i$ , we have to prove the following theorem.

Theorem 2

When $N$ is $C^{\infty }$ and $\parallel N^n({\mu }_0)\parallel \le M\le e^{-1}$ , for all $n$ , the series $\sum _{n=0}^{\infty }{G}_n$ is absolutely convergent.

Proof

Let the relation

$${\varsigma }_n={\varsigma }_0\exp \left({\varsigma }_{n-1}\right)$$

(14)

for $n=1,2,\dots$ .

where ${\varsigma }_0=M.$ Let ${\kappa }_n={\varsigma }_n-{\varsigma }_{n-1}$ . Using ${\mu }_n=G_{n-1}$ and $G_n=\sum _{i_{n=1}}^{\infty }\sum _{i_{n-1=0}}^{\infty }\dots \sum _{i_{1=0}}^{\infty }[N^{(\sum _{k=1}^n{i}_k)}({\mu }_0)\underset{j=1}{\overset{n}{\Pi }}{\mu }_j^{\mathrm{ij}}/i_j!]$ , we get

$$\parallel G_n\parallel \le {\kappa }_n,n=1,2,\dots$$

(15)

Let

$${\omega }_n=\sum _{i=1}^n{\kappa }_i={\varsigma }_n-{\varsigma }_0$$

(16)

Note that ${\varsigma }_0=e^{-1}>0,{\varsigma }_1={\varsigma }_0\exp ({\varsigma }_0)>{\varsigma }_0$ , and ${\varsigma }_2={\varsigma }_0\exp ({\varsigma }_1)={\varsigma }_1$ . Generally, ${\varsigma }_n>{\varsigma }_{n-1}>0$ . Therefore, $\sum {\kappa }_n$ is a series of positive real numbers. Note that

$$\begin{array}{c}0<{\varsigma }_0=M=e^{-1}<1\hfill \\ 0<{\varsigma }_1={\varsigma }_0\exp \left({\varsigma }_0\right)<{\varsigma }_0{e}^1=e^{-1}{e}^1=1\hfill \\ 0<{\varsigma }_2={\varsigma }_0\exp \left({\varsigma }_1\right)<{\varsigma }_0{e}^1=e^{-1}{e}^1=1\hfill \end{array}$$

(17)

Generally, $0<{\varsigma }_n<1$ . So, ${\omega }_n={\varsigma }_n-{\varsigma }_0<1$ , which proves that ${\left\{{\omega }_n\right\}}_{n=1}^{\infty }$ is bounded above by $1$ and hence convergent. Therefore, by comparison test, $\sum G_n$ is absolutely convergent.

Proposed method and materials

Consider the problem

$$L\left(\mu \left(x\right)\right)+N\left(\mu \left(x\right)\right)+G\left(x\right)=0,x\in \mathrm{\Omega }$$

(18)

$$\beta (\mu \left(x\right),\frac{d\mu \left(x\right)}{\mathrm{dx}})=0,x\in T$$

(19)

where $L$ denotes linear operator while $N$ nonlinear. $G(x)$ is the given function. Construct a homotopy, $H(\omega (x,\ell ,c_i)):\mathrm{\Omega }\times [0,1]\to R$ , by

$$\begin{array}{c}H\left(\omega (x,\ell ,c_i)\right)=(1-\ell )[L\left(\omega (x,\ell )\right)-L\left(u_{\mathrm{ini}}\left(x\right)\right)]\hfill \\ +\ell [L\left(\omega (x,\ell )\right)+N(\omega (x,\ell )+G\left(x\right))]=0\hfill \end{array}$$

(20)

where $\ell \in [0,1]$ and presents an embedding parameter, $\omega (x,\ell )$ is an unknown function, and $u_{\mathrm{ini}}(x)$ is an initial approximation for the solution of equation (18), which satisfies the boundary conditions. Clearly, when $\ell =0$ and $\ell =1$ , equation (20) holds and takes the forms as

$$H(\omega ,0)=L\left(\omega (x,0)\right)-L\left(u_{\mathrm{ini}}\left(x\right)\right)=0$$

(21)

$$H(\omega ,1)=L\left(\omega (x,1)\right)+N\left(\omega (x,1)\right)+G\left(x\right)=0$$

(22)

The paths $L(\omega (x,0))-L({\mu }_{\mathrm{ini}}(x))$ and $L(\omega (x,1))+N(\omega (x,1))+G(x)$ are homotopic to each other. For the solution of equations (21) and (22), assume the perturbation series

$$\omega (x,\ell )={\mu }_0\left(x\right)+\ell {\mu }_1\left(x\right)+{\ell }^2{\mu }_2\left(x\right)+\cdots$$

(23)

When $\ell =0$ , then $\omega (x,0)={\mu }_0(x)$ and when $\ell =1$ , then $\omega (x,1)=\overline{\mu }(x)={\mu }_0(x)+{\mu }_1(x)+{\mu }_2(x)+\cdots$ , which is the essence of He’s HPM.

Now, for MOHPM, the nonlinear function $N(\omega (x,\ell ))$ is decomposed as

$$\begin{array}{c}N\left(\omega (x,\ell )\right)=N\left({\mu }_0\right)+\ell [N({\mu }_0+{\mu }_1)-N\left({\mu }_0\right)]\hfill \\ +{\ell }^2[N({\mu }_0+{\mu }_1+{\mu }_2)-N({\mu }_0+{\mu }_1)]+\cdots \hfill \end{array}$$

(24)

The expressions $N({\mu }_0),[N({\mu }_0+{\mu }_1)-N({\mu }_0)],[N({\mu }_0+{\mu }_1+{\mu }_2)-N({\mu }_0+{\mu }_1)],\dots$ on the right-hand side in equation (24) are Daftardar-Gejji and Jafari polynomials.^26,27 For simplicity and convenience, these polynomials are expressed as

$$N_0=N\left({\mu }_0\right),N_m=N\left(\sum _{i=0}^m{\mu }_i\right)-N\left(\sum _{i=0}^{m-1}{\mu }_i\right)$$

We can now express

$$N\left(\omega (x,\ell )\right)=N_0+\sum _{k=1}^{\infty }{\ell }^k{N}_k$$

(25)

Substituting equation (25) into equation (20), and also by introducing a number of unknown auxiliary functions, ${\gamma }_0(x,c_i),{\gamma }_1(x,c_j),{\gamma }_2(x,c_l),$ for $i,j,l=0,1,2,3,\dots$ , that depend on the variable $x$ and some constants $c_0,c_1,c_2,\dots$ , we get a new homotopy as

$$\begin{array}{l}H\left(\omega ,\ell \right)=\left(1-\ell \right)\left[L\left(\omega \left(x,\ell \right)\right)-L\left(u_{ini}\left(x\right)\right)\right]\\ +\ell [L\left(\omega \left(x,\ell \right)\right)+{\gamma }_0\left(x,c_i\right)N_0+{\gamma }_1\left(x,c_j\right)\ell N_1\\ +{\ell }^2{\gamma }_2\left(x,c_l\right)N_2+\cdots +G\left(x\right)]=0\end{array}$$

(26)

Now, comparing the coefficients of similar powers of $\ell$ in equation (26), we get linear differential equations of zeroth order, first order, second order, and so on, which can be solved very easily:

• Zeroth-order problem

$$L\left({\mu }_0\left(x\right)\right)=L\left({\mu }_{\mathrm{ini}}\left(x\right)\right),\beta ({\mu }_0\left(x\right),\frac{d{\mu }_0\left(x\right)}{\mathrm{dx}})=0$$

(27)

• First-order problem

$$\begin{array}{c}L\left({\mu }_1\left(x\right)\right)+{\gamma }_0(x,c_i)\left(N_0\right)+G\left(x\right)=0,\hfill \\ \beta ({\mu }_1\left(x\right),\frac{d{\mu }_1\left(x\right)}{\mathrm{dx}})=0\hfill \end{array}$$

(28)

• Second-order problem

$$L\left({\mu }_2\left(x\right)\right)+{\gamma }_1(x,c_j)\left(N_1\right)=0,\beta ({\mu }_2\left(x\right),\frac{d{\mu }_2\left(x\right)}{\mathrm{dx}})=0$$

(29)

• Third-order problem

$$L\left({\mu }_3\left(x\right)\right)+{\gamma }_2(x,c_l)\left(N_2\right)=0,\beta ({\mu }_3\left(x\right),\frac{d{\mu }_3\left(x\right)}{\mathrm{dx}})=0$$

(30)

where ${\gamma }_0(x,c_i),{\gamma }_1(x,c_j),{\gamma }_2(x,c_l),\dots$ for $i,j,l=0,1,2,3,\dots$ , are not unique and can be chosen as the same form of nonlinear operator $N$ . ¹⁵ We can make higher order problems, but solutions up to the third-order problems are enough to get more accurate results.

Let $\overline{\mu }(x)$ be the mth-order approximate solution, then

$$\overline{\mu }\left(x\right)={\mu }_0\left(x\right)+{\mu }_1\left(x\right)+{\mu }_2\left(x\right)+{\mu }_3\left(x\right)+\cdots +{\mu }_m\left(x\right)$$

(31)

This depends upon the auxiliary functions ${\gamma }_0(x,c_i),{\gamma }_1(x,c_j),{\gamma }_2(x,c_l),\dots$ for $i,j,l=0,1,2,3,\dots$ . The constants $c_0,c_1,c_2,\dots$ which are present in the expressions of auxiliary functions can be determined by different methods such as Galerkin’s method, collocation method, and the least square method. The residual is achieved by substituting $\overline{\mu }(x)$ in equation (18)

$$\begin{array}{cc}\multicolumn{1}{c}{R(x,c_0,c_1,c_2,\dots )=}& L\left(\overline{\mu }(x,c_0,c_1,c_2,\dots )\right)\hfill \\ \multicolumn{1}{c}{}& +N\left(\overline{\mu }(x,c_0,c_1,c_2,\dots )\right)+G\left(x\right)\hfill \end{array}$$

(32)

If $R=0$ , then the solution will be exact. Otherwise, it is minimized by different methods mentioned in Marinca and Herisanu ¹⁴ and Ali et al. ¹⁵ The values of $c_0,c_1,c_2,\dots$ can be calculated by various techniques mentioned above. Here, least square method is used for this purpose as given below.

Defined the function as

$$\zeta (c_0,c_1,c_2,\dots )=\underset{a}{\stackrel{b}{\int }}\left(R{(x,c_0,c_1,c_2,\dots )}^2\right)\mathrm{dx}$$

(33)

and then by minimizing it, we have

$$\frac{\partial \zeta }{\partial c_0}=\frac{\partial \zeta }{\partial c_1}=\frac{\partial \zeta }{\partial c_2}=0\cdots$$

(34)

In Galerekin’s method, the following system is used for finding the auxiliary constants as

$$\underset{a}{\stackrel{b}{\int }}R\frac{\partial \overline{\mu }\left(x\right)}{\partial c_0}\mathrm{dx}=0,\underset{a}{\stackrel{b}{\int }}R\frac{\partial \overline{\mu }\left(x\right)}{\partial c_1}\mathrm{dx}=0,\underset{a}{\stackrel{b}{\int }}R\frac{\partial \overline{\mu }\left(x\right)}{\partial c_2}\mathrm{dx}=0,\dots$$

(35)

Application of MOHPM

In this section, high accuracy of MOHPM is shown over the existing methods in the literature. The proposed method is applied to some nonlinear differential equations of different orders. As a result, we see that MOHPM gives best approximation and takes very less time to produce better results.

Example 1

Fourth-order nonlinear boundary value problem ²⁸

$$\begin{array}{c}\frac{d^4\mu \left(s\right)}{d{s}^4}+\mu (s)\frac{d\mu \left(s\right)}{\mathrm{ds}}-4s^7-24=0,0<s<1\hfill \end{array}$$

(36)

$$\begin{array}{c}\mu \left(0\right)=0,\mu ″\prime (0.25)=6,\mu ″(0.5)=3,\mu \left(1\right)=1\hfill \end{array}$$

(37)

This problem has the following exact solution

$$\begin{array}{c}\mu \left(s\right)=s^4\hfill \end{array}$$

(38)

Applying MOHPM, we obtained the approximate solution of zeroth-order problem for $a=0$ and $b=1$

$$\begin{array}{c}\mu \left(s\right)=s^4\hfill \end{array}$$

(39)

In OHPM, use the following second-order approximate solution

$$\begin{array}{c}\mu \left(s\right)={\mu }_0\left(s\right)+{\mu }_1\left(s\right)+{\mu }_2(s,c_1,c_2)\hfill \end{array}$$

(40)

Applying Galerekin’s method to obtain the following values of c’s for $a=0$ and $b=1$

$$\begin{array}{c}{c}_1=1.000006890,c_2=1.267783870\hfill \end{array}$$

(41)

and the solution in this case is given as

$$\begin{array}{cc}\multicolumn{1}{c}{\mu =}& 6.8537\times {10}^{-8}s+1.9974\times {10}^{-9}{s}^2\hfill \\ \multicolumn{1}{c}{}& +2.7805\times {10}^{-10}{s}^3+s^4\hfill \\ \multicolumn{1}{c}{}& +4.5111\times {10}^{-10}{s}^5+5.16508\times {10}^{-11}{s}^6\hfill \\ \multicolumn{1}{c}{}& +2.3444\times {10}^{-12}{s}^7\hfill \\ \multicolumn{1}{c}{}& -1.6801\times {10}^{-8}{s}^8+1.6923\times {10}^{-8}{s}^9\hfill \\ \multicolumn{1}{c}{}& +5.6160\times {10}^{-10}{s}^{10}\hfill \\ \multicolumn{1}{c}{}& -3.4800\times {10}^{-9}{s}^{11}+7.8966\times {10}^{-13}{s}^{12}\hfill \\ \multicolumn{1}{c}{}& +7.2051\times {10}^{-14}{s}^{13}\hfill \\ \multicolumn{1}{c}{}& +2.8405\times {10}^{-15}{s}^{14}\hfill \end{array}$$

(42)

Results of solutions MOHPM and OHPM are shown in Figure 1 and Table 1.

Figure 1.

Comparison of MOHPM solution and exact solution for Example 1 (Table 1).

Table 1.

Comparison of the errors obtained by MOHPM, OHPM, and OHAM ²⁸ solutions with the exact solution. We observe that the results of our modified method are better and accurate than the other methods in practice.

s	Exact	MOHPM	$E^{*}(\mathrm{MOHPM})$	$E^{*}(\mathrm{OHPM})$	$E^{*}(\mathrm{OHAM})$
0.0	0.00	0.00	0.0	0.0	0.00
0.1	0.0001	0.0001	0.0	6.9E−09	4.6E−18
0.2	0.0016	0.0016	0.0	1.4E−08	8.0E−18
0.3	0.0081	0.0081	0.0	2.1E−08	1.0E−17
0.4	0.0256	0.0256	0.0	2.8E−08	1.1E−17
0.5	0.0625	0.0625	0.0	3.5E−08	1.1E−17
0.6	0.1296	0.1296	0.0	4.2E−08	1.0E−17
0.7	0.2401	0.2401	0.0	4.9E−08	7.9E−18
0.8	0.4096	0.4096	0.0	5.6E−08	4.8E−18
0.9	0.6561	0.6561	0.0	6.2E−08	7.2E−19
1.0	1.00	1.00	0.0	6.9E−08	4.2E−18

MOHPM: modified optimal homotopy perturbation method; OHPM: optimal homotopy perturbation method; OHAM: optimal homotopy asymptotic method.

Example 2

Fifth-order nonlinear boundary value problem ¹⁵

$$\begin{array}{c}\frac{d^5\mu }{d{s}^5}-{\mu }^2{e}^{-s}=0,0<s<1\hfill \end{array}$$

(43)

$$\begin{array}{c}\mu \left(0\right)=1,\mu \prime \left(0\right)=1,\mu ″\left(0\right)=1,\mu \left(1\right)=e,\mu \prime \left(1\right)=e\hfill \end{array}$$

(44)

For this problem, the exact solution is

$$\begin{array}{c}\mu \left(s\right)=e^s\hfill \end{array}$$

(45)

Assume the following second-order approximation, to solve this problem

$$\begin{array}{c}\mu \left(s\right)={\mu }_0\left(s\right)+{\mu }_1(s,c_1)+{\mu }_2(s,c_1,c_2)\hfill \end{array}$$

(46)

Applying Galerkin’s technique to get the values of c_i’s for $a=0$ and $b=1$

$$\begin{array}{c}{c}_1=0.999999240,c_2=0.999758960\hfill \end{array}$$

(47)

Applying MOHPM, the solution is given as

$$\begin{array}{cc}\multicolumn{1}{c}{\mu =}& 1+s+\frac{s^2}{2}+0.166666667s^3+0.041666667s^4\hfill \\ \multicolumn{1}{c}{}& +\frac{s^5}{120}+\frac{s^6}{720}+\frac{s^7}{5040}\hfill \\ \multicolumn{1}{c}{}& +0.000024802s^8+0.000002756s^9+2.7557\times {10}^{-7}\hfill \\ \multicolumn{1}{c}{}& +2.5050\times {10}^{-8}{s}^{11}+2.0928\times {10}^{-9}{s}^{12}\hfill \\ \multicolumn{1}{c}{}& +1.5506\times {10}^{-10}{s}^{-13}+1.4690\times {10}^{-11}{s}^{14}\hfill \end{array}$$

(48)

Applying OHPM, we obtained

$$\begin{array}{c}{c}_1=1.000553563,c_2=-0.417820368\hfill \end{array}$$

(49)

In this case, solution is

$$\begin{array}{cc}\multicolumn{1}{c}{\mu \left(s\right)=}& 1+s+\frac{s^2}{2}+0.166668432s^3+0.041661159s^4\hfill \\ \multicolumn{1}{c}{}& +0.008337946s^5+0.001389658s^6\hfill \\ \multicolumn{1}{c}{}& +0.000198523s^7+0.000019825s^8\hfill \\ \multicolumn{1}{c}{}& +0.000006842s^9-5.0613\times {10}^{-7}{s}^{10}\hfill \\ \multicolumn{1}{c}{}& -4.1383\times {10}^{-8}{s}^{11}\hfill \\ \multicolumn{1}{c}{}& -1.6480\times {10}^{-8}{s}^{12}+1.4653\times {10}^{-8}{s}^{13}\hfill \\ \multicolumn{1}{c}{}& -9.8279\times {10}^{-9}{s}^{14}\hfill \end{array}$$

(50)

The results of solutions MOHPM and OHPM are shown in Table 2 and Figure 2.

Table 2.

Comparison of the errors obtained by MOHPM, OHPM, and OHAM ¹⁵ solutions with the exact solution. We observe that the results of our modified method are better and accurate than the other methods in practice.

s	Exact	MOHPM	$E^{*}(\mathrm{MOHPM})$	$E^{*}(\mathrm{OHPM})$	$E^{*}(\mathrm{OHAM})$
0.0	1.000000000	1.000000000	0.0	0.0	0.0
0.1	1.105170918	1.105170918	3.1E−15	1.2E−09	9.2E−10
0.2	1.221402759	1.221402758	1.9E−14	6.8E−09	5.0E−09
0.3	1.349858808	1.349858808	5.4E−14	1.4E−08	1.1E−08
0.4	1.491824698	1.491824698	1.0E−13	2.0E−08	1.5E−08
0.5	1.648721271	1.648721271	1.4E−13	2.1E−08	1.6E−08
0.6	1.822118800	1.822118800	1.6E−13	1.7E−08	1.4E−08
0.7	2.013752707	2.013752707	1.5E−13	1.2E−08	9.9E−08
0.8	2.225540928	2.225540928	9.9E−14	7.0E−09	5.6E−09
0.9	2.459603111	2.459603111	1.1E−14	2.0E−09	1.0E−09
1.0	2.718281828	2.718281828	1.0E−13	3.0E−09	4.2E−09

MOHPM: modified optimal homotopy perturbation method; OHPM: optimal homotopy perturbation method; OHAM: optimal homotopy asymptotic method.

Figure 2.

Comparison of MOHPM solution and exact solution for Example 2 (Table 2).

Example 3

Sixth-order nonlinear boundary value problem: ¹⁵

Consider a nonlinear problem of order six

$$\begin{array}{c}\frac{d^6\mu }{d{s}^6}-{\mu }^2{e}^{-s}=0,0<s<1\hfill \end{array}$$

(51)

with boundary conditions

$$\begin{array}{l}\mu \left(0\right)=1,\mu \left(1\right)=e,{\mu }^{″}\left(0\right)=-1,\\ {\mu }^{″}\left(1\right)=e,{{\mu }^{‴}}^{\prime }\left(0\right)=1,{{\mu }^{‴}}^{\prime }\left(1\right)=e\end{array}$$

(52)

The exact solution is

$$\begin{array}{c}\mu \left(s\right)=e^s\hfill \end{array}$$

(53)

Consider the second-order approximation

$$\begin{array}{c}\mu \left(s\right)={\mu }_0\left(s\right)+{\mu }_1(s,c_1)+{\mu }_2(s,c_1,c_2)\hfill \end{array}$$

(54)

Applying Galerkin’s method, we get the following values of c_i’s for $a=0$ and $b=1$

$$\begin{array}{c}{c}_1=0.999985968,c_2=0.953594089\hfill \end{array}$$

(55)

The MOHPM solution in this case becomes

$$\begin{array}{cc}\multicolumn{1}{c}{\mu \left(s\right)=}& 1+1.000000000s+\frac{s^2}{2}+0.166666667s^3+\frac{s^4}{24}\hfill \\ \multicolumn{1}{c}{}& +0.008333333s^5+\frac{s^6}{720}\hfill \\ \multicolumn{1}{c}{}& +0.000198413s^7+0.000024801s^8\hfill \\ \multicolumn{1}{c}{}& +0.000002756s^9+2.7561\times {10}^{-7}{s}^{10}\hfill \\ \multicolumn{1}{c}{}& +2.5034\times {10}^{-8}{s}^{11}\hfill \\ \multicolumn{1}{c}{}& +2.0790\times {10}^{-9}{s}^{12}+1.6767\hfill \\ \multicolumn{1}{c}{}& \times {10}^{-10}{s}^{13}+1.1824\times {10}^{-11}{s}^{14}\hfill \end{array}$$

(56)

Applying OHPM, we obtained

$$\begin{array}{c}{c}_1=0.999986482,c_2=0.991801518\hfill \end{array}$$

(57)

Approximate solution is given by

$$\begin{array}{cc}\multicolumn{1}{c}{\mu \left(s\right)=}& 1+1.000000013s+\frac{s^2}{2}+0.166666647s^3+\frac{s^4}{24}\hfill \\ \multicolumn{1}{c}{}& +0.008333348s^5+0.001388870s^6\hfill \\ \multicolumn{1}{c}{}& +0.000198427s^7+0.000024799s^8\hfill \\ \multicolumn{1}{c}{}& +0.000002754s^9+2.7645\times {10}^{-7}{s}^{10}\hfill \\ \multicolumn{1}{c}{}& +2.4830\times {10}^{-8}{s}^{11}\hfill \\ \multicolumn{1}{c}{}& +1.8527\times {10}^{-9}{s}^{12}+3.2686\times {10}^{-10}{s}^{13}\hfill \\ \multicolumn{1}{c}{}& +2.0269\times {10}^{-11}{s}^{14}\hfill \end{array}$$

(58)

Results of solutions MOHPM and OHPM are given in Table 3 and Figure 3.

Table 3.

Comparison of the errors obtained by MOHPM, OHPM, and OHAM ¹⁵ solutions with the exact solution. We observe that the results of our modified method are better and accurate than the other methods in practice.

s	Exact	MOHPM	$E^{*}(\mathrm{MOHPM})$	$E^{*}(\mathrm{OHPM})$	$E^{*}(\mathrm{OHAM})$
0.0	1.	1.	0.	0.	0.
0.1	1.105170918	1.105170918	7.4E−12	1.2E−09	9.2E−10
0.2	1.221402758	1.221402758	1.4E−11	2.4E−09	5.0E−09
0.3	1.349858808	1.349858808	1.9E−11	3.4E−09	1.1E−08
0.4	1.491824698	1.491824698	2.2E−11	4.0E−09	1.5E−08
0.5	1.648721271	1.648721271	2.3E−11	4.3E−09	1.6E−08
0.6	1.822118800	1.822118800	2.2E−11	4.2E−09	1.4E−08
0.7	2.013752707	2.013752707	1.9E−11	3.6E−09	9.9E−09
0.8	2.225540928	2.225540929	1.3E−11	2.6E−09	5.6E−09
0.9	2.459603111	2.459603111	7.2E−12	1.4E−09	1.0E−09
1.0	2.718281828	2.718281828	2.6E−13	2.4E−11	4.2E−09

MOHPM: modified optimal homotopy perturbation method; OHPM: optimal homotopy perturbation method; OHAM: optimal homotopy asymptotic method.

Figure 3.

Comparison of MOHPM solution and exact solution for Example 3 (Table 3).

Example 4

Eighth-order nonlinear boundary value problem: ¹⁶

Consider nonlinear problem

$$\begin{array}{c}\frac{d^8\mu }{d{s}^8}-{\mu }^2{e}^{-s}=0,0<s<1\hfill \end{array}$$

(59)

with boundary conditions

$$\begin{array}{l}\mu \left(0\right)=1,{\mu }^{\prime }\left(0\right)=1,{\mu }^{″}\left(0\right)=1,{\mu }^{‴}\left(0\right)=1,\\ \mu \left(1\right)=e,{\mu }^{\prime }\left(1\right)=e,{\mu }^{″}\left(1\right)=e,{\mu }^{‴}\left(1\right)=e\end{array}$$

(60)

The exact solution is

$$\begin{array}{c}\mu \left(s\right)=e^s\hfill \end{array}$$

(61)

Consider

$$\begin{array}{c}\mu \left(s\right)={\mu }_0\left(s\right)+{\mu }_1(s,c_1)+{\mu }_2(s,c_1,c_2)\hfill \end{array}$$

(62)

By Galerkin’s method, we get the following values of c_i’s for $a=0$ and for $b=1$

$$\begin{array}{c}{c}_1=0.999999394,c_2=2.064518104\hfill \end{array}$$

(63)

The MOHPM solution becomes

$$\begin{array}{cc}\multicolumn{1}{c}{\mu =}& 1+s+\frac{s^2}{2}+\frac{s^3}{6}+0.041666667s^4\hfill \\ \multicolumn{1}{c}{}& +0.008333333s^5+0.001388889s^6\hfill \\ \multicolumn{1}{c}{}& +0.000198413s^7+0.000024802s^8\hfill \\ \multicolumn{1}{c}{}& +0.000002756s^9+2.7557\times {10}^{-7}{s}^{10}\hfill \\ \multicolumn{1}{c}{}& +2.5052\times {10}^{-8}{s}^{11}\hfill \\ \multicolumn{1}{c}{}& +2.0919\times {10}^{-9}{s}^{12}+1.5433\times {10}^{-10}{s}^{13}\hfill \\ \multicolumn{1}{c}{}& +1.5317\times {10}^{-11}{s}^{14}\hfill \end{array}$$

(64)

Applying OHPM and Galerkin’s method, we obtained

$$\begin{array}{c}{c}_1=0.00,c_2=1.9163\times {10}^{-45}\hfill \end{array}$$

(65)

In this case, the solution is

$$\begin{array}{c}\mu =1+1.s+0.5s^2+0.166666667s^3+0.041627502s^4\hfill \\ \multicolumn{1}{c}{+0.008485834s^5+0.001170129s^6+0.000331697s^7}\\ \multicolumn{1}{c}{+0.s^8+0.s^9+0.s^{10}+0.s^{11}+0.s^{12}+0.s^{13}+0.s^{14}}\end{array}$$

(66)

The results of solutions MOHPM and OHPM are shown in Table 4 and Figure 4.

Table 4.

Comparison of the errors obtained by MOHPM, OHPM, and OHAM ¹⁶ solutions with the exact solution. We observe that the results of our modified method are better and accurate than the other methods in practice.

s	Exact	MOHPM	$E^{*}(\mathrm{MOHPM})$	$E^{*}(\mathrm{OHPM})$	$E^{*}(\mathrm{OHAM})$
0.0	1.	1.	0.0	0.0	0.0
0.1	1.105170918	1.105170918	1.1E−15	2.6E−09	2.60E−9
0.2	1.221402758	1.221402758	1.1E−14	2.6E−08	2.62E−8
0.3	1.349858808	1.349858808	3.5E−14	7.9E−08	7.86E−8
0.4	1.491824698	1.491824698	6.1E−14	1.4E−07	1.36E−7
0.5	1.648721271	1.648721271	7.3E−14	1.6E−07	1.61E−7
0.6	1.822118800	1.822118800	6.2E−14	1.4E−07	1.39E−7
0.7	2.013752707	2.013752707	4.0E−14	8.2E−08	8.22E−8
0.8	2.225540928	2.225540929	4.9E−15	2.8E−08	2.80E−8
0.9	2.459603111	2.459603111	3.9E−14	2.8E−09	2.84E−9
1.0	2.718281828	2.718281828	1.7E−13	1.8E−15	1.14E−13

MOHPM: modified optimal homotopy perturbation method; OHPM: optimal homotopy perturbation method; OHAM: optimal homotopy asymptotic method.

Figure 4.

Comparison of MOHPM solution and exact solution for Example 4 (Table 4).

Results and discussion

Here, a new analytical method is created, which gives better results as compared to the results obtained by the existing methods like OHPM and OHAM. Here, Daftardar-Gejji and Jafari polynomials are used in the proposed method for the expansion of nonlinear term in the differential equation. It is economical in terms of computer power/memory and does not involve tedious calculations This method has less limitations and can be applied to every type of differential equation like ODEs, PDEs, IDEs, and their systems. Here, the proposed method is applied to four nonlinear boundary value problems of fourth, fifth, sixth, and eighth orders, and the results are authenticated by comparing with the results obtained by the other methods mentioned in this article. Table 1 and Figures 1 and 2 indicate the solution of fourth-order differential equation. Also, comparison of the errors obtained by MOHPM, OHPM, and OHAM with the exact solution is given in this table. It is observed that the results of the modified method are better. The solution of different boundary value problems of fifth and sixth orders as well as the comparison of the errors with other mentioned methods are explained in Tables 2 and 3, and Figures 3 and 4, which claim the potential of the suggested method. Table 4 shows the solution of nonlinear boundary value problem of eighth order. The achieved errors are also compared with the errors gained by other methods mentioned in Table 4. It is observed that the proposed method is more powerful for the solution of nonlinear boundary value problems of any order.

Conclusion

Here, a new idea (Daftardar-Gejji and Jafari polynomials) has been established and successfully applied this to four nonlinear boundary value problems of fourth, fifth, sixth, and eighth order which provide more accurate results than the standard OHPM and OHAM. The solutions of the MOHPM show outstanding consensus with the exact solution. The quality of MOHPM is that it needs only few steps to achieve good results. It has less limitation. Auxiliary functions, initial guess, optimal convergence control parameters, Galerkin’s method, and least square have great effect on the solution and increase the accuracy. Various ODEs of any order can be solved by this technique. It can also solve PDEs, IDEs, and their systems. This technique has found to be more beneficial for engineers, scientists, and researchers of every field to use.