In this article, a new technique is used to solve the nonlinear boundary value problem of a cantilever-type nanoelectromechanical system. The technique is called the optimal perturbation iteration method and it is used to solve the problem in the form of a nonlinear differential equation with negative power-law nonlinearity. A convergence and error estimation of the proposed method is presented proving that the method is convergent. Results for the application of the proposed technique are demonstrated through two examples and the tables and figures prove that the method is efficient and straightforward.

1 INTRODUCTION

The beam-type electrostatic actuator system has become one the most important system that can be used as an essential component in designing a micro-electromechanical system known as MEMS model and nanoelectromechanical system known as NEMS model.¹ This beam is constructed from a conductive electrode suspended over a conductive substrate. When a voltage difference is noticed between the two electrodes, the upper movable electrode deflects toward the lower one. When this voltage reaches its critical value this upper electrode becomes unstable and pulls on the substrate. Voltage and deflection are some of the parameters that take the name pull-in parameters for the actuators in this situation. The behaviors of the MEMS actuators have been studied over the years to better understand their dynamics. Some brief illustrations for this issue can be found in References 2, 3. The intermolecular and quantum forces were neglected in these studies based on a physical scale. With the development of the use of nanotechnology and the extent of their use in different areas of science, an extensive ongoing search from researchers around the world is being conducted to understand more their performance. If the separation between two molecules is less than 20 nm, then it will be considered as intermolecular Van der Waals attraction.^4-6 On the other hand, if the separation between is more than 20 nm, the force between the two surfaces can have the name of the quantum Camier effect.^7-11 A mathematical model of this problem can be described in the following form

\frac{d^{4} ζ}{d x^{4}} = \frac{α_{K}}{{(1 - ζ (x))}^{K}} + \frac{β}{{(1 - ζ (x))}^{2}} + \frac{γ}{(1 - ζ (x))}

()

with boundary conditions

ζ (0) = ζ^{'} (0) = ζ^{''} (1) = ζ^{'''} (1) = 0; 0 < x < 1 .

()

Over the past few years, there have been some considerable studies for acquiring the solution for the model (37) of this problem. However, very few works are found for solving it. For example, Duan et al.¹² used a modified version of the adomian method called the Duan Rach adomian decomposition method for solving the problem. In Reference 12, the problem is converted in a Fredholm–Volterra integral equation and then this converted system is solved using the modified adomian method. Also, in Reference 13 the homotopy perturbation method is applied for solving this model. An approximation for the effect of different parameters of the Casimir force on the instability of electrostatic nanocantilevers is discussed in Reference 14. Due to the lack of works on solving this problem, the motivation of this article arises.

Homotopy perturbation method (HAM) and the optimal perturbation iteration method (OPIM) gained increasing interest recently with their ability to provide accurate solutions to different forms of similar problems. The method and other related techniques have been tested multiple times on models with application in science, engineering, biology, and other related applied fields. For instance, this method has been adapted for solving the Jeffery–Hamel flow problem which has great application in fluid mechanics.¹⁵ Also, the approximate solution for the squeezing flow between two plates is investigated using the same methods. Other important models including system of nonlinear coupled differential equations occurring in the phenomenon of fluid,^{16, 17} nonconservative dynamical system of a rotating electrical machine,¹⁸ strongly nonlinear equation,^19-21 delay differential equation,²² nonlinear heat transfer equations,²³ Lane-Emden-type equations,^{24, 25} Fisher's equations,²⁶ Bratu-type problems,²⁷ generalized regularized long wave equations,²⁸ Burgers's equation,²⁹ modified Boussinesq-Burger equations,³⁰ coupled Drinfel'd-Sokolov-Wilson equation,³¹ and Klein-Gordon equations.³² Other models that have been solved using this method can be found in References 33-37 and references therein.

In this article, we are interested in investigating the effect of different parameters for solving the modem problem and their approximate solution. The organization of the article is as follows: Section 2 introduces the main steps of the OPIA method for solving the model (37). In Section 3 a detailed convergence analysis for the proposed technique is presented along with an error estimation for the solution. In Section 4, results for solving model (37) with the OPIA method are demonstrated through tables and figures proving the effectiveness of the method. Finally, Section 5 gives a conclusion of the study.

2 OPTIMAL PERTURBATION ITERATION METHOD

In this section, we shall provide the main steps for the optimal homotopy method. First, considering Equation (37), one can rewrite the main problem in a closed form as:

F (ζ^{(4)}, ζ, ε) = 0,

()

where

ζ = ζ (x)

and

ε

is the perturbation parameter. The optimal perturbation iteration algorithms (OPIAs) can be obtained by using the idea of perturbation theory and Taylor's theorem. One can take the approximate solution with one correction term from straightforward perturbation expansion as

ζ_{n + 1} = ζ_{n} + ε {(ζ_{c})}_{n},

()

where

n \in N \cup \{0\}

and

{(ζ_{c})}_{n}

is the nth correction term of the iteration algorithm. Upon substitution of (4) into (3) then expanding it in a Taylor series with nth derivatives yields the OPIA-n s. Taking only first derivatives from the Taylor's expansion, we have OPIA-1 as

F + F_{ζ {(ζ_{c})}_{n}} ε + F_{ζ^{(4)}} {(ζ_{c}^{(4)})}_{n} ε + F_{ε} ε = 0,

()

where subscripts of F denotes partial differentiation and all derivatives and functions are computed at

ε = 0

. We can reformulate the above algorithm as follows:

{(ζ_{c}^{(4)})}_{n} + \frac{F_{ζ}}{F_{ζ^{(4)}}} {(ζ_{c})}_{n} = - \frac{\frac{F}{ε} + F_{ε}}{F_{ζ^{(4)}}} .

()

An initial function

ζ_{0}

satisfying the prescribed condition(s) must be selected to obtain the first correction term from the following algorithm:

{(ζ_{c}^{(4)})}_{0} + \frac{F_{ζ}}{F_{ζ^{(4)}}} {(ζ_{c})}_{0} = - \frac{\frac{F}{ε} + F_{ε}}{F_{ζ^{(4)}}} .

()

One can use the above equation to get the approximate results in the desired limits. To start the iteration procedure, a first trial function

ζ_{0}

is selected appropriately according to the prescribed conditions. The first correction term

{(ζ_{c})}_{0}

can be computed from the algorithms (36) by using

u_{0}

and given condition(s). Then the first approximate solution

ζ_{1}

is obtained by using

{(ζ_{c})}_{0}

and so on. To get better and more effective approximations, we propose a new approach to these algorithms. Based on the idea of HAM,^38-40 we insert a convergence-control parameters

P_{0}, P_{1}, P_{2}, \dots

into Equation (4) and then construct new components, defined by

\begin{array}{l} ζ_{1} (x; P_{0}) & = ζ_{0} + P_{0} {(ζ_{c})}_{0}, \\ ζ_{2} (x; P_{1}) & = ζ_{1} + P_{1} {(ζ_{c})}_{1}, \\ ⋮ \\ ζ_{m} (x; P_{m - 1}) & = ζ_{m - 1} + P_{m - 1} {(ζ_{c})}_{m - 1} . \end{array}

()

In order to obtain the optimum values of these paramters, we make use of the similar strategy mentioned by Marinca et al.⁴¹ where the method of least squares or collocation techniques are used. Substituting the approximate solution

ζ_{m}

into Equation (3), we will get the following residual:

R (x, P_{0}, \dots, P_{m - 1}) = F ({(ζ_{m})}^{(4)}, ζ_{m}) .

()

It is clear that, when

R (x, P_{0}, \dots, P_{m - 1}) = 0

then the approximation

ζ_{m}

is the exact solution of the problems. Generally such case will not arise for nonlinear equations, but one can minimize the functional

J (P_{0}, \dots, P_{m - 1}) = \int_{a}^{b} R^{2} (x, P_{0}, \dots, P_{m - 1}) d x,

()

where a and b are elected from the domain of the problem. Optimum values of

P_{0}, P_{1}, \dots

can be optimally defined from the conditions

J_{P_{0}} = J_{P_{1}} = \dots = J_{P_{m - 1}}

. OPIM has been applied to many types of nonlinear equation. Some of them can be seen in References 42-46

3 CONVERGENCE ANALYSIS AND ERROR ESTIMATE

In this section, we will investigate the convergence of the proposed optimal perturbation iteration technique with the aid of some theorems. New approximate solution obtained by OPIM are considered as follows:

D_{0} = ζ_{0}, D_{n + 1} = P_{n} {(ζ_{c})}_{n} .

()

Correspondingly, other OPIM solutions can be determined as:

\begin{array}{l} ζ_{0} & = D_{0}, \\ ζ_{1} & = ζ_{0} + P_{0} {(ζ_{c})}_{0} = D_{0} + D_{1}, \\ ζ_{2} & = ζ_{1} + P_{1} {(ζ_{c})}_{1} = D_{0} + D_{1} + D_{2}, \\ ζ_{3} & = ζ_{2} + P_{2} {(ζ_{c})}_{2} = D_{0} + D_{1} + D_{2} + D_{3}, \\ ⋮ \\ ζ_{n + 1} & = ζ_{n} + P_{n} {(ζ_{c})}_{n} = D_{0} + D_{1} + D_{2} + \dots + D_{n + 1} = \sum_{i = 0}^{n + 1} D_{i} . \end{array}

()

Therefore, one can represent the approximate solution of the problem as:

ζ (x) = \lim_{n \to \infty} ζ_{n + 1} (x) = \sum_{i = 0}^{\infty} D_{i} .

()

Theorem 1.Let us assume that B denotes a Banach space and

A : B \to B

()

is a nonlinear mapping and also we suppose that

‖A [y] - A [\overline{y}]‖ \leq β ‖y - \overline{y}‖, y, \overline{y} \in B,

()

for

0 < β < 1

, where

β

is some constant. Then, the mapping A has a unique fixed point. Additionally, the following sequence

ζ_{n + 1} = A [ζ_{n}],

()

with an arbitrary selection of

ζ_{0} \in B

, converges to the fixed point of the mapping A and

‖ζ_{r} - ζ_{s}‖ \leq ‖ζ_{1} - ζ_{0}‖ \sum_{j = s - 1}^{r - 2} β^{j} .

()

Banach fixed point theorem may be used to derive the following theorem.

Theorem 2.Let B represents a Banach space designated with an appropriate norm $‖.‖$ over which the series $\sum_{i = 0}^{\infty} D_{i}$ is defined and let us assume that the initial mapping $ζ_{0} = D_{0}$ falls inside the ball of the exact solution $ζ (x)$ . So, the solution $\sum_{i = 0}^{\infty} D_{i}$ converges if there is a $β$ such that

‖D_{n + 1}‖ \leq β ‖D_{n}‖,

()

where

0 < β < 1

Proof.To prove the above theorem, let us first define a sequence as:

\begin{array}{l} A_{0} & = D_{0}, \\ A_{1} & = D_{0} + D_{1}, \\ A_{2} & = D_{0} + D_{1} + D_{2}, \\ ⋮ \\ A_{n} & = D_{0} + D_{1} + D_{2} + \dots + D_{n} . \end{array}

()

We must show that ${\{A_{n}\}}_{n = 0}^{\infty}$ is a Cauchy sequence in B. In order to achieve that, we consider

‖A_{n + 1} - A_{n}‖ = ‖ζ_{n + 1}‖ \leq β ‖ζ_{n}‖ \leq β^{2} ‖ζ_{n - 1}‖ \leq \dots \leq β^{n + 1} ‖D_{0}‖ .

()

For every $n, k \in N, n \geq k$ , we have

\begin{array}{l} ‖A_{n} - A_{k}‖ & = ‖(A_{n} - A_{n - 1}) + (A_{n - 1} - A_{n - 2}) + \dots + (A_{k + 1} - A_{k})‖ \\ \leq ‖A_{n} - A_{n - 1}‖ + ‖A_{n - 1} - A_{n - 2}‖ + \dots + ‖A_{k + 1} - A_{k}‖ \\ \leq β^{n} ‖D_{0}‖ + β^{n - 1} ‖D_{0}‖ + \dots + β^{k + 1} ‖D_{0}‖ = \frac{1 - β^{n - k}}{1 - β} β^{k + 1} ‖D_{0}‖ . \end{array}

()

Since it is known that $0 < β < 1$ , one can easily get from (21)

\lim_{n, k \to \infty} ‖A_{n} - A_{k}‖ = 0 .

()

Finally, ${\{A_{n}\}}_{n = 0}^{\infty}$ is a Cauchy sequence in B and this implies that the series solution (12) is convergent. This completes the proof.

Theorem 3.If $ζ_{0} = D_{0}$ falls inside the ball of the solution $ζ (x)$ , then $A_{n} = \sum_{i = 0}^{n} D_{i}$ remains inside that ball, too.

Proof.Assume that

D_{0} \in B_{r} (ζ),

()

where

B_{r} (ζ) = \{D \in A| ‖ζ - D‖ < r\}

()

is the ball of

D (x)

. From the hypothesis

ζ = \lim_{n \to \infty} A_{n} = \sum_{i = 0}^{\infty} D_{i}

and using Theorem 2, we get

‖ζ - A_{n}‖ \leq β^{n + 1} ‖D_{0}‖ < ‖D_{0}‖ < r,

()

where

β \in (0, 1)

and

n \in N

Theorem 4.Let us now suppose that $\sum_{i = 0}^{\infty} D_{i}$ , that is, the approximate OPIM solution, is convergent to the desired solution $ζ (x)$ . If the truncated series $\sum_{i = 0}^{k} D_{i}$ is utilized as an approximation to the (3), we obtain

E_{k} (x) \leq \frac{β^{k + 1}}{1 - β} ‖D_{0}‖,

()

where the

E_{k}

is the maximum error.

Proof.By using Equation (21), one can get

‖A_{n} - A_{k}‖ \leq \frac{1 - β^{n - k}}{1 - β} β^{k + 1} ‖D_{0}‖

()

for

n \geq k

. By knowing

ζ (x) = \lim_{n \to \infty} A_{n} (x) = \sum_{i = 0}^{\infty} D_{i}

()

one can write

‖ζ (x) - \sum_{i = 0}^{k} D_{i}‖ \leq \frac{1 - β^{n - k}}{1 - β} β^{k + 1} ‖D_{0}‖

()

and also it can be rewritten as

E_{k} (x) = ‖ζ (x) - \sum_{i = 0}^{k} D_{i}‖ \leq \frac{β^{k + 1}}{1 - β} ‖D_{0}‖

()

since

1 - β^{n - k} < 1

. Here

β

is chosen as

β = \max \{β_{i}, i = 0, 1, \dots, n\}

where

β_{i} = \frac{‖D_{n + 1}‖}{‖D_{n}‖} .

()

4 NUMERICAL EXPERIMENTS

In this section, we try to find new approximate solutions to modified equation by using perturbation algorithms through two illustrative examples. Firstly, Equation (37) with perturbation parameter can be written as :

\begin{array}{l} F (ζ^{(4)}, ζ, ε) \\ = ζ^{(4)} - \frac{ε α_{K}}{{(1 - ζ (x))}^{K}} - \frac{ε β}{{(1 - ζ (x))}^{2}} - \frac{γ}{(1 - ζ (x))} . \end{array}

()

Taking the approximate solutions as

ζ_{n + 1} = ζ_{n} + ε {(ζ_{c})}_{n}

()

and putting it into the algorithm

F + F_{ζ {(ζ_{c})}_{n}} ε + F_{ζ^{(4)}} {(ζ_{c}^{(4)})}_{n} ε + F_{ε} ε = 0

()

one can get

{(ζ_{c}^{(4)})}_{0} + \frac{F_{ζ}}{F_{ζ^{(4)}}} {(ζ_{c})}_{0} = - \frac{\frac{F}{ε} + F_{ε}}{F_{ζ^{(4)}}}

()

or explicitly

\begin{array}{l} {(ζ_{c}^{(4)})}_{0} - (K {(1 - ζ)}^{- K - 1} - \frac{γ}{{(1 - ζ)}^{2}} + \frac{2 β}{{(1 - ζ)}^{3}}) \\ = - \frac{α_{K}}{{(1 - ζ (x))}^{K}} - \frac{β}{{(1 - ζ (x))}^{2}} - \frac{γ}{(1 - ζ (x))} \end{array}

()

by setting

ε = 1

after computations. In order to see the efficiency of the proposed technique, one can start to iterations with an initial function which satisfies the boundary conditions.

Experiment 1

In the lights of above settings, let us consider

\frac{d^{4} ζ}{d x^{4}} = \frac{α_{K}}{{(1 - ζ (x))}^{K}} + \frac{β}{{(1 - ζ (x))}^{2}} + \frac{γ}{(1 - ζ (x))}

()

with boundary conditions

ζ (0) = ζ^{'} (0) = ζ^{''} (1) = ζ^{'''} (1) = 0; 0 < x < 1 .

()

In order to initialize the iterations, one needs to choose the initial function which must satisfy the following conditions

ζ (0) = ζ^{'} (0) = ζ^{''} (1) = ζ^{'''} (1) = 0

()

such as

ζ (x) = \frac{1}{24} (x^{4} - 4 x^{3} + 6 x^{2}) .

Therefore, the first-order approximate solution will be in the following form:

ζ_{1} = \frac{P_{0}}{24} (- x^{4} β - x^{4} γ + 24 x^{2} c_{3} + 24 x^{3} c_{4} - x^{4} α_{K}) .

()

With the solution (40) and proceeding as in Section 3, second-order approximate solution can be obtained as:

\begin{array}{l} ζ_{2} = ζ_{1} + \frac{143 P_{1} K}{3072} \\ \times (\begin{array}{l} \frac{β^{2} c_{4} x^{11}}{95040} + \frac{β γ c_{4} x^{11}}{95040} + \frac{K α_{K} β^{2} c_{3} x^{10}}{60480} + \frac{β γ c_{3} x^{10}}{60480} - \frac{β c_{3} c_{4} x^{9}}{1512} - \frac{β c_{3}^{2} x^{8}}{1680} \\ + \frac{1}{840} γ c_{4} K α_{K} x^{7} + \frac{1}{180} β c_{3} x^{6} + \frac{1}{360} γ c_{3} x^{6} - \frac{K α_{K} β^{3} x^{12}}{6842880} \\ - \frac{K β^{2} γ x^{12}}{3421440} - \frac{β^{2} x^{12} α_{K}}{855360} - \frac{β^{3} x^{16} α_{K}}{150958080} - \frac{β γ^{2} x^{12}}{6842880} - \frac{K β^{2} x^{8}}{20160} - \frac{β γ x^{8}}{13440} - \frac{γ^{2} x^{8}}{40320} \\ - \frac{β x^{4}}{24} - \frac{K γ x^{4}}{24} - \frac{β^{4} x^{20} α_{K}}{38578913280} - \frac{β^{2} γ x^{16} α_{K}}{50319360} - \frac{β γ^{2} x^{16} α_{K}}{50319360} - \frac{β^{3} γ x^{20} α_{K}}{9644728320} \\ - \frac{7 K β γ x^{12} α_{K}}{3421440} - \frac{γ^{2} x^{12} α_{K}}{1140480} - \frac{β x^{8} α_{K}}{6720} - \frac{γ x^{8} α_{K}}{8064} + \frac{x^{4} α_{K}}{24} - \frac{β γ α_{K}^{3} x^{20}}{3214909440} \\ - \frac{β γ^{2} α_{K}^{2} x^{20}}{3214909440} - \frac{β^{2} γ α_{K}^{2} x^{20}}{3214909440} - \frac{β^{2} α_{K}^{3} x^{20}}{6429818880} - \frac{γ^{2} α_{K}^{3} x^{20}}{6429818880} - \frac{β α_{K}^{4} x^{20}}{9644728320} \\ - \frac{β^{3} α_{K}^{2} x^{20}}{9644728320} - \frac{γ^{3} α_{K}^{2} x^{20}}{9644728320} - \frac{α_{K}^{5} x^{20}}{38578913280} + \frac{β c_{3} α_{K}^{3} x^{18}}{84602880} + \frac{c_{3} α_{K}^{4} x^{18}}{253808640} \\ + \frac{c_{4} α_{K}^{4} x^{19}}{321490944} + \frac{β c_{4} α_{K}^{3} x^{19}}{107163648} + \frac{γ c_{3} α_{K}^{3} x^{18}}{84602880} - \frac{γ α_{K}^{4} x^{20}}{9644728320} + \frac{β γ c_{4} α_{K}^{2} x^{19}}{53581824} + \\ \frac{γ c_{4} α_{K}^{3} x^{19}}{107163648} + \frac{β^{2} c_{4} α_{K}^{2} x^{19}}{107163648} + \frac{γ^{2} c_{4} α_{K}^{2} x^{19}}{107163648} + \dots \end{array}) . \end{array}

()

\begin{array}{l} ζ_{3} = ζ_{2} + 8053 P_{2} x^{4} \\ \times (\begin{array}{l} \frac{β^{2} c_{3} K^{2} α_{K}^{2} x^{18}}{84602880} + \frac{γ^{2} c_{3} α_{K}^{2} x^{18}}{84602880} + \frac{K (K + 1) β γ c_{3} α_{K}^{2} x^{18}}{42301440} - \frac{β c_{4}^{2} α_{K}^{2} x^{18}}{3525120} \\ - \frac{γ c_{4}^{2} α_{K}^{2} x^{18}}{3525120} - \frac{c_{4}^{2} α_{K}^{3} x^{18}}{7050240} + \frac{K (K + 2) c_{4}^{3} α_{K}^{2} x^{17}}{342720} - \frac{β c_{3} c_{4} α_{K}^{2} x^{17}}{1370880} - \\ \frac{γ c_{3} c_{4} α_{K}^{2} x^{17}}{1370880} - \frac{c_{3} c_{4} α_{K}^{3} x^{17}}{2741760} + \frac{c_{3} c_{4}^{2} α_{K}^{2} x^{16}}{87360} - \frac{c_{4}^{4} α_{K} x^{16}}{43680} \\ - \frac{β c_{3}^{2} α_{K}^{2} x^{16}}{2096640} - \frac{γ c_{3}^{2} α_{K}^{2} x^{16}}{2096640} - \frac{c_{3}^{2} α_{K}^{3} x^{16}}{4193280} - \frac{β γ α_{K}^{2} x^{16}}{25159680} \\ - \frac{β α_{K}^{3} x^{16}}{50319360} - \frac{γ α_{K}^{3} x^{16}}{50319360} - \frac{β^{2} α_{K}^{2} x^{16}}{50319360} - \frac{γ^{2} α_{K}^{2} x^{16}}{50319360} - \frac{α_{K}^{4} x^{16}}{150958080} + \\ \frac{c_{4} α_{K}^{3} x^{15}}{1572480} + \frac{c_{3}^{2} c_{4} α_{K}^{2} x^{15}}{65520} + \frac{K (K + 1) β c_{4} α_{K}^{2} x^{15}}{786240} + \frac{γ c_{4} α_{K}^{2} x^{15}}{786240} + \frac{c_{3} c_{4}^{3} α_{K} x^{15}}{8190} \\ + \frac{c_{3} α_{K}^{3} x^{14}}{1153152} + \frac{K^{3} c_{3}^{3} α_{K}^{2} x^{14}}{144144} + \frac{β γ c_{4} x^{19} α_{K}^{2}}{53581824} + \frac{γ^{2} c_{4} x^{19} α_{K}^{2}}{107163648} - \\ \frac{β c_{3} α_{K}^{2} x^{14}}{576576} + \frac{γ c_{3} α_{K}^{2} x^{14}}{576576} - \frac{c_{4}^{2} α_{K}^{2} x^{14}}{48048} - \frac{c_{3} c_{4} α_{K}^{2} x^{13}}{17160} - \frac{c_{3}^{2} α_{K}^{2} x^{12}}{23760} \\ - \frac{γ α_{K}^{2} x^{12}}{570240} - \frac{α_{K}^{3} x^{12}}{1140480} - \frac{13 β α_{K}^{2} x^{12}}{6842880} + \frac{c_{4} α_{K}^{2} x^{11}}{15840} + \frac{c_{3} α_{K}^{2} x^{10}}{10080} - \frac{α_{K}^{2} x^{8}}{10080} \\ - \frac{β^{2} c_{3} c_{4} x^{17} α_{K}}{2741760} - \frac{β γ c_{3} c_{4} x^{17} α_{K}}{1370880} - \frac{β c_{3} c_{4} x^{13} α_{K}}{17160} - \frac{γ c_{3} c_{4} x^{13} α_{K}}{17160} + \dots \end{array}) \end{array}

()

and so on. It should be mentioned that integration constants $c_{3}, c_{4}$ in the first-order solution can be obtained by using the other boundary conditions. For instance, the values for the third-order OPIM approximations are $c_{3} = - 0.50338, c_{4} = 1.08906$ . By using these values and proceeding as in Section 2, one can get desired approximate OPIM solutions. Optimal values of convergence parameters are given in Tables 1 and 2. For different values and parameters, absolute errors between the numerical and analytical results can be seen in Tables 3–6. From Figures 1 and 2, one can see the absolute residual errors of third- and fourth-order OPIM approximations for $β = 0.25$ , $β = 0.35$ , and $β = 0.45$ where $K = 3, γ = 0.6, α_{K} = 0.2$ . From those results, one can conclude that the more we iterate, the better results we have.

Details are in the caption following the image — **FIGURE 1**
Open in figure viewer PowerPoint

Absolute residual errors of third-order OPIM approximation for $β = 0.25$ ( $■$ ), $β = 0.35$ ( $▴$ ), and $β = 0.45$ ( $•$ ) where $K = 3, γ = 0.6, α_{K} = 0.2$

TABLE 1. OPIM constants for different values of parameters for third-order solutions

	$K = 3, α_{K} = 0.2,$ $β = 0.6, γ = 0.25$	$K = 3, α_{K} = 0.25,$ $β = 0.8, γ = 0.3$
$P_{0}$	1.023617	2.55214
$P_{1}$	4.063215	$- 1.30215$
$P_{2}$	$- 3.90588$	$- 0.83694$

TABLE 2. OPIM constants for different values of parameters for fourth-order solutions and for

K = 3

	$α_{K} = 0.2, β = 0.6, γ = 0.25$	$α_{K} = 0.2, β = 0.5, γ = 0.25$	$α_{K} = 0.3, β = 0.4, γ = 0.25$
$P_{0}$	1.33844	0.40283	$- 0.05638$
$P_{1}$	1.50447	1.08827	1.50842
$P_{2}$	$- 0.99063$	$- 1.70244$	$- 0.80478$
$P_{3}$	1.04727	0.80228	1.77014

TABLE 3. Absolute residual errors of the third-order OPIM solutions for

K = 3

and

β = 0.55

x	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$3.445 \times 10^{- 5}$	$3.945 \times 10^{- 5}$	$7.116 \times 10^{- 6}$	$6.551 \times 10^{- 6}$
0.2	$1.378 \times 10^{- 3}$	$3.156 \times 10^{- 4}$	$7.112 \times 10^{- 5}$	$1.288 \times 10^{- 5}$
0.3	$3.101 \times 10^{- 3}$	$1.065 \times 10^{- 3}$	$3.601 \times 10^{- 4}$	$1.350 \times 10^{- 4}$
0.4	$5.512 \times 10^{- 3}$	$2.525 \times 10^{- 3}$	$1.138 \times 10^{- 3}$	$5.691 \times 10^{- 4}$
0.5	$8.613 \times 10^{- 3}$	$4.931 \times 10^{- 3}$	$2.778 \times 10^{- 3}$	$1.737 \times 10^{- 4}$
0.6	$1.240 \times 10^{- 2}$	$8.522 \times 10^{- 3}$	$5.761 \times 10^{- 3}$	$4.321 \times 10^{- 4}$
0.7	$1.688 \times 10^{- 2}$	$1.353 \times 10^{- 2}$	$1.067 \times 10^{- 2}$	$9.340 \times 10^{- 4}$
0.8	$2.205 \times 10^{- 2}$	$2.020 \times 10^{- 2}$	$1.821 \times 10^{- 2}$	$1.821 \times 10^{- 3}$
0.9	$2.791 \times 10^{- 2}$	$2.876 \times 10^{- 2}$	$2.916 \times 10^{- 2}$	$3.281 \times 10^{- 3}$
1.0	$3.445 \times 10^{- 2}$	$3.945 \times 10^{- 2}$	$4.445 \times 10^{- 2}$	$5.557 \times 10^{- 3}$

TABLE 4. Absolute residual errors of the third-order OPIM solutions for

K = 3

and

β = 0.75

x	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$3.106 \times 10^{- 6}$	$8.054 \times 10^{- 7}$	$4.111 \times 10^{- 6}$	$9.777 \times 10^{- 6}$
0.2	$5.741 \times 10^{- 4}$	$4.208 \times 10^{- 5}$	$1.044 \times 10^{- 6}$	$9.011 \times 10^{- 6}$
0.3	$1.937 \times 10^{- 3}$	$2.130 \times 10^{- 4}$	$3.086 \times 10^{- 5}$	$1.006 \times 10^{- 5}$
0.4	$4.593 \times 10^{- 3}$	$6.733 \times 10^{- 4}$	$1.301 \times 10^{- 4}$	$1.770 \times 10^{- 5}$
0.5	$8.969 \times 10^{- 3}$	$1.644 \times 10^{- 3}$	$3.969 \times 10^{- 4}$	$6.753 \times 10^{- 5}$
0.6	$1.550 \times 10^{- 2}$	$3.408 \times 10^{- 3}$	$9.876 \times 10^{- 4}$	$2.017 \times 10^{- 4}$
0.7	$2.461 \times 10^{- 2}$	$6.315 \times 10^{- 3}$	$2.135 \times 10^{- 3}$	$5.085 \times 10^{- 4}$
0.8	$3.674 \times 10^{- 2}$	$1.077 \times 10^{- 2}$	$4.162 \times 10^{- 3}$	$1.133 \times 10^{- 3}$
0.9	$5.231 \times 10^{- 2}$	$1.726 \times 10^{- 2}$	$7.499 \times 10^{- 3}$	$2.297 \times 10^{- 3}$
1.0	$7.176 \times 10^{- 2}$	$2.631 \times 10^{- 2}$	$1.270 \times 10^{- 2}$	$4.322 \times 10^{- 3}$

TABLE 5. Absolute residual errors of the third-order OPIM solutions for

K = 4

and

β = 0.25

$x$	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$1.057 \times 10^{- 7}$	$5.201 \times 10^{- 6}$	$1.991 \times 10^{- 6}$	$1.044 \times 10^{- 6}$
0.2	$7.991 \times 10^{- 6}$	$9.111 \times 10^{- 6}$	$7.771 \times 10^{- 6}$	$9.012 \times 10^{- 6}$
0.3	$6.806 \times 10^{- 4}$	$3.897 \times 10^{- 6}$	$1.271 \times 10^{- 5}$	$3.241 \times 10^{- 5}$
0.4	$1.613 \times 10^{- 3}$	$3.955 \times 10^{- 7}$	$5.355 \times 10^{- 5}$	$1.821 \times 10^{- 4}$
0.5	$3.151 \times 10^{- 3}$	$2.947 \times 10^{- 4}$	$1.634 \times 10^{- 4}$	$6.945 \times 10^{- 4}$
0.6	$5.445 \times 10^{- 3}$	$1.521 \times 10^{- 3}$	$4.066 \times 10^{- 4}$	$2.074 \times 10^{- 3}$
0.7	$8.646 \times 10^{- 3}$	$6.089 \times 10^{- 3}$	$8.789 \times 10^{- 4}$	$5.229 \times 10^{- 3}$
0.8	$1.291 \times 10^{- 2}$	$2.025 \times 10^{- 2}$	$1.714 \times 10^{- 3}$	$1.165 \times 10^{- 2}$
0.9	$1.838 \times 10^{- 2}$	$5.846 \times 10^{- 2}$	$3.088 \times 10^{- 3}$	$2.362 \times 10^{- 2}$
1.0	$2.521 \times 10^{- 2}$	$1.509 \times 10^{- 1}$	$5.229 \times 10^{- 3}$	$4.445 \times 10^{- 2}$

TABLE 6. Absolute residual errors of the third-order OPIM solutions for

K = 4

and

β = 0.45

x	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$2.362 \times 10^{- 4}$	$1.525 \times 10^{- 4}$	$4.673 \times 10^{- 3}$	$3.357 \times 10^{- 4}$
0.2	$6.047 \times 10^{- 3}$	$1.952 \times 10^{- 3}$	$2.991 \times 10^{- 3}$	$4.297 \times 10^{- 3}$
0.3	$5.499 \times 10^{- 3}$	$3.336 \times 10^{- 3}$	$3.407 \times 10^{- 3}$	$7.341 \times 10^{- 4}$
0.5	$1.142 \times 10^{- 3}$	$1.204 \times 10^{- 4}$	$7.354 \times 10^{- 4}$	$2.626 \times 10^{- 4}$
0.6	$4.911 \times 10^{- 3}$	$4.314 \times 10^{- 4}$	$2.196 \times 10^{- 3}$	$9.408 \times 10^{- 4}$
0.7	$1.685 \times 10^{- 3}$	$3.232 \times 10^{- 4}$	$1.234 \times 10^{- 3}$	$7.048 \times 10^{- 3}$
0.9	$1.259 \times 10^{- 3}$	$7.370 \times 10^{- 3}$	$2.501 \times 10^{- 3}$	$1.608 \times 10^{- 3}$
1.0	$2.924 \times 10^{- 3}$	$1.541 \times 10^{- 3}$	$4.707 \times 10^{- 3}$	$3.361 \times 10^{- 3}$

Experiment 2

Next, let us now consider the main equation represented in Equation (37) with the new setting of boundary conditions in the form

ζ (0) = 1, ζ^{'} (0) = ζ^{''} (1) = ζ^{'''} (1) = 0,

()

with the initial function can be chosen as

ζ (x) = \frac{1}{24 + 6 x^{2} - 4 x^{3} + x^{4}},

which is suitable for the conditions

ζ (0) = 1, ζ^{'} (0) = ζ^{''} (1) = ζ^{'''} (1) = 0 .

()

By doing the similar computations as in the previous example, one can reach the desired approximations. Optimal values of convergence parameters are given in Tables 7 and 8. For different values and parameters, absolute errors between the numerical and analytical results can be seen in Tables 9–12. From Figures 1 and 2, one can see the absolute residual errors of third- and fourth-order OPIM approximations for $β = 0.25$ , $β = 0.35$ , and $β = 0.45$ where $K = 3, γ = 0.6, α_{K} = 0.2$ . We can get better solutions by taking more terms from the iterations.

TABLE 7. Experiment 2: OPIM constants for different values of parameters for third-order solutions

	$K = 3, α_{K} = 0.35,$ $β = 0.75, γ = 0.45$	$K = 3, α_{K} = 0.3,$ $β = 0.8, γ = 0.35$
$P_{0}$	0.80563	1.800963
$P_{1}$	2.00963	$- 0.990532$
$P_{2}$	$- 2.056381$	1.020116

TABLE 8. Experiment 2: OPIM constants for different values of parameters for fourth-order solutions and for

K = 3

	$α_{K} = 0.35, β = 0.75, γ = 0.35$	$α_{K} = 0.4, β = 0.65, γ = 0.25$	$α_{K} = 0.3, β = 0.4, γ = 0.25$
$P_{0}$	2.00124	1.10458	0.00238
$P_{1}$	0.83775	2.04580	3.77455
$P_{2}$	$- 0.39688$	0.004528	$- 5.01445$
$P_{3}$	2.11057	1.048891	6.05895

TABLE 9. Experiment 2: Absolute residual errors of the third-order OPIM solutions for

K = 3

and

β = 0.55

x	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$8.8733 \times 1 0^{- 4}$	$1.9616 \times 1 0^{- 2}$	$2.9424 \times 1 0^{- 2}$	$3.9232 \times 1 0^{- 1}$
0.2	$1.4907 \times 1 0^{- 3}$	$4.6314 \times 1 0^{- 3}$	$6.9471 \times 1 0^{- 3}$	$9.2628 \times 1 0^{- 2}$
0.3	$1.7042 \times 1 0^{- 3}$	$1.8686 \times 1 0^{- 3}$	$2.8029 \times 1 0^{- 3}$	$3.7372 \times 1 0^{- 2}$
0.4	$1.3982 \times 1 0^{- 3}$	$9.1129 \times 1 0^{- 4}$	$1.3669 \times 1 0^{- 3}$	$1.8226 \times 1 0^{- 2}$
0.5	$4.1926 \times 1 0^{- 4}$	$4.7638 \times 1 0^{- 4}$	$7.1457 \times 1 0^{- 4}$	$9.5277 \times 1 0^{- 3}$
0.6	$1.4096 \times 1 0^{- 3}$	$2.4736 \times 1 0^{- 4}$	$3.7105 \times 1 0^{- 4}$	$4.9473 \times 1 0^{- 3}$
0.7	$4.2888 \times 1 0^{- 3}$	$1.1584 \times 1 0^{- 4}$	$1.7375 \times 1 0^{- 4}$	$2.3167 \times 1 0^{- 3}$
0.8	$8.4424 \times 1 0^{- 3}$	$3.6569 \times 1 0^{- 5}$	$5.4854 \times 1 0^{- 5}$	$7.3138 \times 1 0^{- 4}$
0.9	$1.4118 \times 1 0^{- 2}$	$1.2007 \times 1 0^{- 5}$	$1.8011 \times 1 0^{- 5}$	$2.4014 \times 1 0^{- 4}$
1.0	$2.1585 \times 1 0^{- 2}$	$4.1215 \times 1 0^{- 5}$	$6.1823 \times 1 0^{- 5}$	$8.243 \times 1 0^{- 4}$

TABLE 10. Experiment 2: Absolute residual errors of the third-order OPIM solutions for

K = 3

and

β = 0.75

x	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$1.5795 \times 1 0^{- 3}$	$3.5902 \times 1 0^{- 2}$	$2.8404 \times 1 0^{- 2}$	$5.0017 \times 1 0^{- 1}$
0.2	$2.6534 \times 1 0^{- 3}$	$8.4766 \times 1 0^{- 3}$	$6.7063 \times 1 0^{- 3}$	$1.1809 \times 1 0^{- 1}$
0.3	$3.0335 \times 1 0^{- 3}$	$3.42 \times 1 0^{- 3}$	$2.7057 \times 1 0^{- 3}$	$4.7645 \times 1 0^{- 2}$
0.4	$2.4888 \times 1 0^{- 3}$	$1.6679 \times 1 0^{- 3}$	$1.3195 \times 1 0^{- 3}$	$2.3236 \times 1 0^{- 2}$
0.5	$7.4627 \times 1 0^{- 4}$	$8.719 \times 1 0^{- 4}$	$6.898 \times 1 0^{- 4}$	$1.2147 \times 1 0^{- 2}$
0.6	$2.509 \times 1 0^{- 3}$	$4.5274 \times 1 0^{- 4}$	$3.5818 \times 1 0^{- 4}$	$6.3073 \times 1 0^{- 3}$
0.7	$7.6341 \times 1 0^{- 3}$	$2.1201 \times 1 0^{- 4}$	$1.6773 \times 1 0^{- 4}$	$2.9536 \times 1 0^{- 3}$
0.8	$1.5028 \times 1 0^{- 2}$	$6.6931 \times 1 0^{- 5}$	$5.2952 \times 1 0^{- 5}$	$9.3244 \times 1 0^{- 4}$
0.9	$2.513 \times 1 0^{- 2}$	$2.1976 \times 1 0^{- 5}$	$1.7386 \times 1 0^{- 5}$	$3.0616 \times 1 0^{- 4}$
1.0	$3.8422 \times 1 0^{- 2}$	$7.5434 \times 1 0^{- 5}$	$5.9679 \times 1 0^{- 5}$	$1.0509 \times 1 0^{- 3}$

TABLE 11. Experiment 2: Absolute residual errors of the third-order OPIM solutions for

K = 4

and

β = 0.25

$x$	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$8.547 \times 1 0^{- 5}$	$2.9232 \times 1 0^{- 3}$	$3.143 \times 1 0^{- 3}$	$5.9396 \times 1 0^{- 2}$
0.2	$1.4359 \times 1 0^{- 4}$	$6.9017 \times 1 0^{- 4}$	$7.4207 \times 1 0^{- 4}$	$1.4024 \times 1 0^{- 2}$
0.3	$1.6415 \times 1 0^{- 4}$	$2.7846 \times 1 0^{- 4}$	$2.9939 \times 1 0^{- 4}$	$5.6579 \times 1 0^{- 3}$
0.4	$1.3468 \times 1 0^{- 4}$	$1.358 \times 1 0^{- 4}$	$1.4601 \times 1 0^{- 4}$	$2.7593 \times 1 0^{- 3}$
0.5	$4.0384 \times 1 0^{- 5}$	$7.0991 \times 1 0^{- 5}$	$7.6328 \times 1 0^{- 5}$	$1.4424 \times 1 0^{- 3}$
0.6	$1.3577 \times 1 0^{- 4}$	$3.6862 \times 1 0^{- 5}$	$3.9634 \times 1 0^{- 5}$	$7.49 \times 1 0^{- 4}$
0.7	$4.1311 \times 1 0^{- 4}$	$1.7262 \times 1 0^{- 5}$	$1.856 \times 1 0^{- 5}$	$3.5074 \times 1 0^{- 4}$
0.8	$8.1319 \times 1 0^{- 4}$	$5.4495 \times 1 0^{- 6}$	$5.8593 \times 1 0^{- 6}$	$1.1073 \times 1 0^{- 4}$
0.9	$1.3598 \times 1 0^{- 3}$	$1.7893 \times 1 0^{- 6}$	$1.9238 \times 1 0^{- 6}$	$3.6357 \times 1 0^{- 5}$
1.0	$2.0791 \times 1 0^{- 3}$	$6.1419 \times 1 0^{- 6}$	$6.6037 \times 1 0^{- 6}$	$1.248 \times 1 0^{- 4}$

TABLE 12. Experiment 2: Absolute residual errors of the third-order OPIM solutions for

K = 4

and

β = 0.45

$x$	$α_{K} = 0.2, γ = 0.2$	$α_{K} = 0.25, γ = 0.25$	$α_{K} = 0.3, γ = 0.2$	$α_{K} = 0.2, γ = 0.3$
0.1	$1.4483 \times 1 0^{- 6}$	$1.8686 \times 1 0^{- 3}$	$9.8706 \times 1 0^{- 3}$	$7.8353 \times 1 0^{- 6}$
0.2	$2.4331 \times 1 0^{- 6}$	$1.1528 \times 1 0^{- 4}$	$2.4058 \times 1 0^{- 3}$	$6.4627 \times 1 0^{- 4}$
0.3	$2.7816 \times 1 0^{- 6}$	$2.0798 \times 1 0^{- 4}$	$1.0295 \times 1 0^{- 3}$	$7.6398 \times 1 0^{- 4}$
0.4	$2.2821 \times 1 0^{- 6}$	$3.1999 \times 1 0^{- 4}$	$5.526 \times 1 0^{- 4}$	$8.0476 \times 1 0^{- 4}$
0.5	$6.8431 \times 1 0^{- 7}$	$3.7088 \times 1 0^{- 4}$	$3.3595 \times 1 0^{- 4}$	$8.2329 \times 1 0^{- 4}$
0.6	$2.3007 \times 1 0^{- 6}$	$3.9768 \times 1 0^{- 4}$	$2.2186 \times 1 0^{- 4}$	$8.3305 \times 1 0^{- 4}$
0.7	$7.0002 \times 1 0^{- 6}$	$4.1307 \times 1 0^{- 4}$	$1.5634 \times 1 0^{- 4}$	$8.3865 \times 1 0^{- 4}$
0.8	$1.378 \times 1 0^{- 5}$	$4.2234 \times 1 0^{- 4}$	$1.1685 \times 1 0^{- 4}$	$8.4203 \times 1 0^{- 4}$
0.9	$2.3043 \times 1 0^{- 5}$	$4.2802 \times 1 0^{- 4}$	$9.2654 \times 1 0^{- 5}$	$8.441 \times 1 0^{- 4}$
1.0	$3.5232 \times 1 0^{- 5}$	$4.3144 \times 1 0^{- 4}$	$7.8104 \times 1 0^{- 5}$	$8.4534 \times 1 0^{- 4}$

It can be easily deduced that as the number of iterations increase, the approximate solution becomes more tortuous and the use of the symbolic computer program becomes indispensable. Mathematica 9.0 is used to handle with the complex computations for solving this problem.

5 CONCLUSION

In this article, an analytical method is adapted for solving the nonlinear differential equation modeling the nanoelectromagnetic actuator model. This technique is based on a new modified method called the optimal iteration perturbation method. Detailed convergence analysis for the proposed technique is illustrated and the provided results through two numerical experiments confirm the convergence of this method. The graphical representation of the residual error confirms that the method is effective and provides accurate results. It is interesting to investigate the application of this technique for solving some similar nonlinear problems in the future.

ACKNOWLEDGMENT

The authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions which further enriched the paper and improved its scientific contents.

CONFLICT OF INTEREST

The authors declares no conflict of interest.

Biographies

Waleed Adel received the B.Sc. degree civil engineering and the M.Sc. and Ph.D. degrees in engineering mathematics from Mansoura University, Egypt, in 2007, 2014, and 2019, respectively. He works now as an Assistant Professor with the department of mathematics and engineering physics at the Faculty of Engineering, Mansoura University since 2019. He published more than 40 papers in prestigious journals in the field of applied mathematics and a reviewer for several international reputed journals. His current research interests include but not limited to numerical analysis for differential equations, developing novel solutions using spectral methods, computational mathematics, stability analysis for some biological models.
Sinan Deniz was born in Uşak, Turkey, in 1989. He is a research assistant of Mathematics at Manisa Celal Bayar University since September 2012. He received his B.Sc. degree in Mathematics from Fatih University, Istanbul, Turkey. He completed his master study in Applied Mathematics at Manisa Celal Bayar University in 2014. He obtained his Ph.D. in 2018 at Manisa Celal Bayar University with a dissertation about optimization of analytical approximate solutions of ODEs and PDEs. He has more than 40 SCI papers about applied mathematics and physics. He is referee of several international journals in the frame of pure and applied mathematics. His main research interests are nonlinear ordinary and partial differential equations, asymptotic and computational methods for ODEs and PDEs, semianalytical and numerical methods, fractional differential equations, complex function theory, perturbation techniques, computational analysis, mathematical modeling, mathematical physics, integral equations, numerical analysis, and stability theory.

Open Research

DATA AVAILABILITY STATEMENT

All data generated or analyzed during this study are included in this article.

REFERENCES

1Pelesko JA, Bernstein DH. Modeling Mems and Nems. CRC Press; 2002.
10.1201/9781420035292
Google Scholar
2Kuang J-H, Chen C-J. Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators. Math Comput Model. 2005; 41(13): 1479-1491.
10.1016/j.mcm.2005.06.001
Google Scholar
3Lin W, Zhao Y. Pull-in instability of micro-switch actuators: model review. Int J Nonlinear Sci Numer Simul. 2008; 9(2): 175.
10.1515/IJNSNS.2008.9.2.175
Web of Science® Google Scholar
4Israelachvili JN. Intermolecular and Surface Forces. Academic Press; 2011.
Google Scholar
5Kooch A, Abadyan M. Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method. Trends Appl Sci Res. 2012; 7(1): 57.
10.3923/tasr.2012.57.67
Google Scholar
6Ramezani A, Alasty A, Akbari J. Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces. Int J Solids Struct. 2007; 44(14-15): 4925-4941.
10.1016/j.ijsolstr.2006.12.015
CAS Web of Science® Google Scholar
7Mostepanenko VM, Mostepanenko VM, Mostepanenko VM, Trunov NN, Mostepanenko V, RL Z. The Casimir Effect and Its Applications. Oxford University Press; 1997.
10.1093/oso/9780198539988.001.0001
Google Scholar
8Lamoreaux SK. The Casimir force: background, experiments, and applications. Rep Prog Phys. 2004; 68(1): 201.
10.1088/0034-4885/68/1/R04
Web of Science® Google Scholar
9Koochi A, Kazemi AS, Beni YT, Yekrangi A, Abadyan M. Theoretical study of the effect of Casimir attraction on the pull-in behavior of beam-type NEMS using modified Adomian method. Physica E. 2010; 43(2): 625-632.
10.1016/j.physe.2010.10.009
CAS Google Scholar
10Lin W-H, Zhao Y-P. Nonlinear behavior for nanoscale electrostatic actuators with Casimir force. Chaos Soliton Fract. 2005; 23(5): 1777-1785.
10.1016/S0960-0779(04)00442-4
Web of Science® Google Scholar
11Rodriguez AW, Capasso F, Johnson SG. The Casimir effect in microstructured geometries. Nat Photonics. 2011; 5(4): 211.
10.1038/nphoton.2011.39
CAS Web of Science® Google Scholar
12Duan J-S, Rach R, Wazwaz A-M. Solution of the model of beam-type micro-and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems. Int J Non-Linear Mech. 2013; 49: 159-169.
10.1016/j.ijnonlinmec.2012.10.003
Web of Science® Google Scholar
13Mojahedi M, Moghimi Zand M, Ahmadian MT. Static pull-in analysis of electrostatically actuated microbeams using homotopy perturbation method. Appl Math Model. 2010; 34(4): 1032-1041.
10.1016/j.apm.2009.07.013
Web of Science® Google Scholar
14Abadyan M, Novinzadeh A, Kazemi AS. Approximating the effect of the Casimir force on the instability of electrostatic nano-cantilevers. Phys Scr. 2010; 81(1):015801.
10.1088/0031-8949/81/01/015801
Web of Science® Google Scholar
15Ali L, Islam S, Gul T. Modified optimal homotopy perturbation method to investigate Jeffery-Hamel flow. Punjab Univ J Math. 2019; 51(11): 17-29.
Web of Science® Google Scholar
16Ali L, Islam S, Gul T. An improved form of optimal homotopy asymptotic method for the solution of a system of nonlinear coupled differential equations occurring in the phenomenon of fluid mechanics. AIP Conf Proc. 2019; 2116(1): 300006.
10.1063/1.5114306
Google Scholar
17Bildik N, Deniz S. Modification of perturbation-iteration method to solve different types of nonlinear differential equations. AIP Conf Proc. 2017; 1798:020027.
10.1063/1.4972619
Google Scholar
18Herişanu N, Marinca V. Optimal homotopy perturbation method for a non-conservative dynamical system of a rotating electrical machine. Zeitschrift für Naturforschung A. 2012; 67(8-9): 509-516.
10.5560/zna.2012-0047
CAS Google Scholar
19Mabood F. Comparison of optimal homotopy asymptotic method and homotopy perturbation method for strongly non-linear equation. J Assoc Arab Univ Basic Appl Sci. 2014; 16: 21-26.
Google Scholar
20Deniz S, Bildik N. Applications of optimal perturbation iteration method for solving nonlinear differential equations. AIP Conf Proc. 2017; 1798:020046.
10.1063/1.4972638
Google Scholar
21Bildik N, Deniz S. Comparative study between optimal homotopy asymptotic method and perturbation-iteration technique for different types of nonlinear equations. Iranian J Sci Technol Trans A Sci. 2018; 42(2): 647-654.
10.1007/s40995-016-0039-2
Web of Science® Google Scholar
22Bildik N, Deniz S. A new efficient method for solving delay differential equations and a comparison with other methods. Eur Phys J Plus. 2017; 132(1): 51.
10.1140/epjp/i2017-11344-9
Web of Science® Google Scholar
23Deniz S. Optimal perturbation iteration method for solving nonlinear heat transfer equations. J Heat Transf-ASME. 2017; 139( 7):074503-1.
10.1115/1.4036085
Web of Science® Google Scholar
24Deniz S, Bildik N. A new analytical technique for solving Lane-Emden type equations arising in astrophysics. Bull Belgian Math Soc Simon Stevin. 2017; 24(2): 305-320.
10.36045/bbms/1503453712
Web of Science® Google Scholar
25Verma AK, Pandit B, Agarwal RP. On approximate stationary radial solutions for a class of boundary value problems arising in epitaxial growth theory. J Appl Comput Mech. 2020; 6(4): 713-734.
Web of Science® Google Scholar
26Bildik N, Deniz S. A practical method for analytical evaluation of approximate solutions of Fisher's equations. ITM Web Conf. 2017; 13:01001.
10.1051/itmconf/20171301001
Google Scholar
27Deniz S, Bildik N. Optimal perturbation iteration method for Bratu-type problems. J King Saud Univ Sci. 2018; 30(1): 91-99.
10.1016/j.jksus.2016.09.001
Web of Science® Google Scholar
28Bildik N, Deniz S. New analytic approximate solutions to the generalized regularized long wave equations. Bull Korean Math Soc. 2018; 55(3): 749-762.
Web of Science® Google Scholar
29Bildik N, Deniz S. Solving the Burgers and regularized long wave equations using the new perturbation iteration technique. Numer Methods Part Differ Equat. 2018; 34(5): 1489-1501.
10.1002/num.22214
Web of Science® Google Scholar
30Deniz S. Semi-analytical investigation of modified Boussinesq-Burger equations. J BAUN Inst Sci Technol. 2020; 22(1): 327-333.
Google Scholar
31Deniz S. Modification of coupled Drinfel'd-Sokolov-Wilson Equation and approximate solutions by optimal perturbation iteration method. Afyon Kocatepe Univ J Sci Eng. 2020; 20(1): 35-40.
10.35414/akufemubid.649745
Google Scholar
32Bildik N, Deniz S. New approximate solutions to the nonlinear using perturbation iteration techniques. Discr Continu Dyn Syst Ser S. 2020; 13(03): 503-518.
Google Scholar
33Agarwal P, Deniz S, Jain S, Alderremy AA, Aly S. A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques. Phys A Stat Mech Appl. 2020; 542:122769.
10.1016/j.physa.2019.122769
Web of Science® Google Scholar
34Deniz S. Iterative perturbation technique for solving a special magnetohydrodynamics problem. Celal Bayar Univ J Sci. 2020; 16(1): 69-74.
CAS Google Scholar
35 Bildik N, Deniz S, Saad KM. A comparative study on solving fractional cubic isothermal auto-catalytic chemical system via new efficient technique. Chaos Soliton Fract. 2020; 132.
10.1016/j.chaos.2019.109555
Web of Science® Google Scholar
36Deniz S, Konuralp A, De la Sen M. Optimal perturbation iteration method for solving fractional model of damped Burgers' equation. Symmetry-Basel. 2020; 12(6): 958.
10.3390/sym12060958
Web of Science® Google Scholar
37Bildik N, Deniz S. Optimal iterative perturbation technique for solving Jeffery-Hamel flow with high magnetic field and nanoparticle. J Appl Anal Comput. 2020; 10(6): 2476-2490. https://doi.org/10.11948/20190378
Google Scholar
38Liao S. Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC Press; 2003.
10.1201/9780203491164
Google Scholar
39Liao S. On the homotopy analysis method for nonlinear problems. Appl Math Comput. 2004; 147(2): 499-513.
10.1016/S0096-3003(02)00790-7
Web of Science® Google Scholar
40Liao S. Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput. 2005; 169(2): 1186-1194.
10.1016/j.amc.2004.10.058
Web of Science® Google Scholar
41Herişanu N, Marinca V, Dordea T, Madescu G. A new analytical approach to nonlinear vibration of an electrical machine. Proc Rom Acad Ser A. 2008; 9(3): 229-236.
Web of Science® Google Scholar
42Deniz S. Semi-analytical approach for solving a model for HIV infection of CD4⁺ T-cells, TWMS. J Appl Eng Math. 2021; 11(1): 273-281.
Web of Science® Google Scholar
43De la Sen M, Deniz S, Sözen H. A new efficient technique for solving modified Chua's circuit model with a new fractional operator. Adv Differ Equat. 2021; 2021: 21.
10.1186/s13662-020-03175-x
Web of Science® Google Scholar
44Deniz S. Optimal perturbation iteration method for solving fractional FitzHugh-Nagumo equation. Chaos Soliton Fract. 2021; 142: 110417.
10.1016/j.chaos.2020.110417
Web of Science® Google Scholar
45Deniz S. Semi-analytical analysis of Allen-Cahn model with a new fractional derivative. Math Methods Appl Sci. 2021; 44(3): 2355-2363.
10.1002/mma.5892
Web of Science® Google Scholar
46Srivastava HM, Deniz S, Saad KM. An efficient semi-analytical method for solving the generalized regularized long wave equations with a new fractional derivative operator. J King Saud Univ Sci. 2021; 33(2): 101345.
10.1016/j.jksus.2021.101345
Web of Science® Google Scholar

Citing Literature

All articles

Approximate solution of the electrostatic nanocantilever model via optimal perturbation iteration method

Abstract

1 INTRODUCTION

2 OPTIMAL PERTURBATION ITERATION METHOD

3 CONVERGENCE ANALYSIS AND ERROR ESTIMATE

4 NUMERICAL EXPERIMENTS

5 CONCLUSION

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biographies

Open Research

DATA AVAILABILITY STATEMENT

REFERENCES

Citing Literature

Figures

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Approximate solution of the electrostatic nanocantilever model via optimal perturbation iteration method

Abstract

1 INTRODUCTION

2 OPTIMAL PERTURBATION ITERATION METHOD

3 CONVERGENCE ANALYSIS AND ERROR ESTIMATE

4 NUMERICAL EXPERIMENTS

5 CONCLUSION

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biographies

Open Research

DATA AVAILABILITY STATEMENT

REFERENCES

Citing Literature

Figures

References

Related

Information