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In this article, we report for the first time the application of a novel and extremely valuable methodology called the Rach–Adomian–Meyers decomposition method (MDM) to obtain numerical solutions to the rotational pendulum equation. MDM is a tool for solving nonlinear differential equations that combines both series solution and the Adomian decomposition method efficiently. We present a simple and highly accurate MDM-based algorithm and its numerical implementation via a one-step recurrence approach for obtaining periodic solutions to the rotational pendulum equation. Finally, numerical simulations are performed to demonstrate the efficiency and accuracy of the proposed technique for both large and small amplitudes of oscillation.

This paper considers an interesting variation of the modified decomposition method, which permits determination of the solution of nonlinear initial-boundary value problems for second-order ODEs appearing in physics such as the Thomas–Fermi, Bratu and Troesch equations.

The modified decomposition method is applied to analyze the transverse vibrations of tapered rotating beams incorporating both axial centrifugal stiffening and flexible end constraints. Unlike prio...

In the present research, an advanced decomposition technique based on the Adomian decomposition method is proposed to achieve the highly accurate numerical solution of non-linear initial value problems of Bratu’s-type without any linearization, perturbation and discretization. For the completeness of the proposed technique, convergence analysis is also addressed. The reliability, generality and validity of the proposed technique are examined by calculating the absolute errors of some initial value problems of Bratu’s type. Moreover, the obtained solutions are compared graphically with the precise solution and also with some existing approaches solutions.

In the present paper, an efficient modification the convergence parameter based on the Adomian Decomposition Method (ADM) is proposed and investigated for a class of nonlinear evolution equations; specifically, the Korteweg de Vries (KdV) equations. We show that the proposed analysis possesses increased accuracy when compared to the standard ADM. Moreover, the optimal value of such a convergence parameter is determined by minimizing the averaged residual error. For such a convergence parameter value, an approximate solution is found to be closer to the available exact solution than the corresponding approximate solution without a convergence parameter for the same number of solution components. The approach proposed may be readily extended to other nonlinear differential and integral equations.

The existence and uniqueness of the solution is proved for a nonlinear boundary value problem for ODE subject to an infinite condition \cite{1}, which describes the study of two-phase Stefan problems on the semi-infinite line [0,\infty). This result considerably extends the analysis of a recent work \cite{4}. A highly accurate analytic approximate solution of this problem is also provided via the Adomian decomposition method.

In this paper, the Adomian decomposition method (ADM) is proposed to solve the hyperbolic Kepler equation which is often used to describe the eccentric anomaly of a comet of extrasolar origin in its hyperbolic trajectory past the Sun. A convenient method is therefore needed to solve this equation to accurately determine the radial distance and/or the Cartesian coordinates of the comet. It has been shown that Adomian's series using a few terms are sufficient to achieve extremely accurate numerical results even for much higher values of eccentricity than those in the literature. Besides, an exceptionally rapid rate of convergence of the sequence of the obtained approximate solutions has been demonstrated. Such approximate solutions possess the odd property in the mean anomaly which are illustrated through several plots. Moreover, the absolute remainder error, using only three components of Adomian's solution decreases across a specified domain, approaches zero as the eccentric anomaly tends to infinity. Also, the absolute remainder error decreases by increasing the number of components of the Adomian decomposition series. In view of the obtained results, the present method may be the most effective approach to treat the hyperbolic Kepler equation.

The present research is devoted to theoretical study of the pull-in performance of double-sided and paddle-type NEMS actuators fabricated from cylindrical nanowire operating in the Casimir regime and in the presence of the centrifugal force. D'Alembert's principle was used to transform the angular velocity into an equivalent static, centrifugal force. Using the couple stress theory, the constitutive equations of the actuators were derived. The equivalent boundary condition technique was applied to obtain the governing equation of the paddle-type actuator. Three distinct approaches, the Duan-Adomian Method (DAM), Finite Difference Method (FDM), and Lumped Parameter Model (LPM), were applied to solve the equation of motion of these two actuators. This study demonstrates the influence of various parameters, i.e., the Casimir force, geometric characteristics, and the angular speed, on the pull-in performance. (C) 2017 Sharif University of Technology. All rights reserved.

The general form of explicit, analytical solution of the Frost-Kalkwarf equation was developed for the first time by means of the Adomian decomposition method as a reliable mathematical tool. The accuracy of the obtained formulas was further improved by applying the diagonal Pade approximants. We have demonstrated that our developed formulas are at least 30 times faster than the original Frost-Kalkwarf equation in computation of vapour pressures and are highly accurate with an overall absolute relative deviation of 2.25% for 88 different substances over 26 400 experimental data points from triple to critical point temperatures. As another unique advantage, our formulas avoid any divergent behaviour in using iterative methods unlike the implicit Frost-Kalkwarf equation. Based on the conducted statistical analyses, our formulas have excellent predictability of vapour pressure values with a probability of 90.33% for absolute relative errors of less than 5%. This article is protected by copyright. All rights reserved

The temperature-dependent dynamics of a negative temperature coefficient (NTC) thermistor conducting variable electric current is modeled using the differential approach. The thermistor is assumed to follow the Steinhart–Hart resistance-temperature equation. The developed mathematical model consists of a nonlinear differential-algebraic equations system, and it was analyzed by the Adomian decomposition method (ADM) and its time-marching version known as the multistage Adomian decomposition method (MADM) as well as the Dormand–Prince (DP) numerical method. Five sets of experiments were conducted on five different NTC thermistors and the laboratory measurements were compared with the model predictions. It is demonstrated that the proposed model, when combined with the MADM, can accurately simulate the thermal behavior of the NTC thermistors. The MADM reproduces the experimental temperature dynamics of the five NTC thermistors with an average absolute relative error of about 2.601% while the corresponding errors for the DP method and the classic ADM are 8.122% and 51.255%, respectively. Also, it is shown that the MADM is highly efficient in terms of computational efficiency and it is approximately 6.5 times faster than the classic DP method, when tuned appropriately.

This article shows that the well-known nonlinear boundary value problem of the differential equation for temperature distribution of convective straight fins with temperature-dependent thermal conductivity is exactly solvable in an implicit form. Furthermore, an exact solution in an explicit form is derived. Also, an accurate analytic solution (series solution) is obtained by a new variation of the Adomian decomposition method.

Paddle-type and double-sided nanostructures have much potential for measuring the angular speed in rotary systems and aerospace applications. While most investigations in the literature have concentrated on the electromechanical performance of conventional beam-type nanostructures, few researchers have addressed the performance of these systems. The pull-in instability of the cantilever paddle-type and double-sided sensors in the presence of the centrifugal force have been investigated. The nonlinear governing equations are solved and the obtained results are compared with the numerical solution. The influences of the van der Waals force, geometric parameters, angular speed, and size phenomenon on the instability performance have been demonstrated.

Purpose The purpose of this paper is to first deduce a new form of the exact analytic solution of the well-known nonlinear second-order differential equation subject to a set of mixed nonlinear Robin and Neumann boundary conditions that model the thin film flows of fourth-grade fluids, and second to compare the approximate analytic solutions by the Adomian decomposition method (ADM) with the new exact analytic solution to validate its accuracy for parametric simulations of the thin film fluid flows, even for more complex models of non-Newtonian fluids in industrial applications. Design/methodology/approach The approach to calculating a new form of the exact analytic solution of thin film fluid flows rests upon a sequence of transformations including the modification of the classic technique due to Scipione del Ferro and Niccolò Fontana Tartaglia. Next the authors establish a lemma that justifies the new expression of the exact analytic solution for thin film fluid flows of fourth-grade fluids. Second, the authors apply a modification of the systematic ADM to quickly and easily calculate the sequence of analytic approximate solutions for this strongly nonlinear model of thin film flow of fourth-grade fluids. The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific and engineering applications. Herein, the authors seek to establish the relative merits of the ADM in the context of the thin film flows of fourth-grade fluids. Findings The ADM is shown to closely agree with the new expression of the exact analytic solution. The authors have calculated the error remainder functions and the maximal error remainder parameters in the error analysis to corroborate the solutions. The error analysis demonstrates the rapid rate of convergence and that we can approximate the exact solution as closely as we please; furthermore the rate of convergence is shown to be approximately exponential, and thus only a low-stage approximation will be adequate for engineering simulations as previously documented in the literature. Originality/value This paper presents an accurate work for solving thin film flows of fourth-grade fluids. The authors have compared the approximate analytic solutions by the ADM with the new expression of the exact analytic solution for this strongly nonlinear model. The authors commend this technique for more complex thin film fluid flow models.

Two reliable methods, namely the Adomian decomposition method (ADM) and the variational iteration method (VIM), are used for solving the Volterra integral equation with a weakly singular kernel in the reproducing kernel space. Both methods provide convergent series solutions for this equation. The ADM method gives a sequence of components of the solution, which composes a sequence of approximations, whereas the VIM more directly provides a sequence of approximations; both exhibit high accuracy. Four numerical examples are examined to confirm the validity and the power of these two methods.

The U-shaped and double-sided nanostructures are promising for developing miniature angular speed sensors. While the electromechanical instability of conventional beam-type nanostructures has been extensively addressed in the literature, few researchers have investigated this phenomenon in the double-sided and U-shaped sensors. In this regard, the present work demonstrates the effect of the centrifugal force on the pull-in performance of the double-sided and U-shaped sensors fabricated from cylindrical nanowire and operated in the van der Waals (vdW) regime. Based on the modified couple stress theory, the size-dependent constitutive equations of the sensors are derived. The governing equations are solved by two different approaches, i.e. the analytic Duan–Adomian method and the numerical differential quadrature method. The influences of the vdW and centrifugal forces, geometric parameters and the size phenomenon on the pull-in parameters are demonstrated.

The influence of the surface energy on the instability of nano-structures under the electrostatic force has been investigated in recent years by different researchers. It appears that in all prior research, the response of all structures becomes softer due to the surface effects. In the present study, the pull-in instability of a NEMS device incorporating the electrostatic force and Casimir intermolecular attraction for different values of the surface parameter is investigated by the Duan–Rach method of determined coefficients (MDC) in order to identify the remarkable effect of the surface energy. Although the obtained results verify the behavior of such structures in presence of the fringing field and the Casimir attraction same as the previous investigations, however the incremental effects of the surface energy cause the aforementioned structures to behave more stiffly in contrast.

In this paper, we establish higher order numeric solutions for the IVP of the singular Lane–Emden-type equation, including the Emden–Fowler equation. We use the multi-stage modified decomposition method to effectively treat these types of equations and develop numeric solutions that are effective in the large. The step-size and the order in our numeric solutions are two parameters that may be arbitrarily specified. Fast algorithms of the Adomian polynomials guarantee the efficiency of our approach, and a higher order numeric solution can be readily generated at will. The proposed method overcomes the singular behaviour at the origin and exhibits approximations of high accuracy with a large effective region of convergence. Several numerical examples are examined to demonstrate the reliability of our new approach. In these examples, we have demonstrated that our numeric solutions are consistent by halving the step-size, i.e. the numeric solutions of different step-sizes nearly coincide.

Chaos control in the Belousov–Zhabotinsky-CSTR system was investigated theoretically and experimentally by reconstructing the phase space of the cerium (IV) ions concentration time series and then optimizing recurrence quantification analysis measures. The devised feedback loop acting on the reactor inlet flow rate was able to experimentally suppress chaos and drive the system to an almost predictable state with approximately 93% determinism. Similar theoretical results have also been demonstrated in numerical simulations using the four-variable Montanator model as solved by the multistage Adomian decomposition method.

We develop a robust algorithm for isothermal flash calculations of multi-component mixtures. The algorithm provides an explicit, parametric solution for a recent variation of the Rachford–Rice equation by employing a powerful analytical technique, namely the Adomian decomposition method. The computational speed of the algorithm is shown to be further enhanced by applying the iterated Shanks transformation. Unlike previous methods, our algorithm obviates the troublesome need for an initial guess, guarantees an excellent rate of convergence and is demonstrated to be more computationally economic than prior art. Several reliable examples from the literature, including the flash calculation for a mixture with nineteen components, as well as an extensive set of hypothetical, randomly-generated problems are presented to illustrate the exceptional efficacy of our new approach.

A number of control schemes including nonlinear feedback, dislocated feedback, and speed feedback have been proposed and assessed for a bromate-malonic acid-cerium Belousov–Zhabotinsky batch reaction process. The tuning parameters of the Oregonator model were firstly adjusted based on a UV-vis spectrophotometric analysis in the experimental part of the research. The adjusted Oregonator model successfully reproduced the innate induction time and periodicity of the BZ-batch system. Subsequently, the controllers were implemented and numerical simulations were carried out by employing the multi-stage Adomian decomposition method. The nominal analysis method was used to study the linear stability of each design. All the controlled systems were found to be linearly stable for certain continuous regions of controller gain. The performance of the proposed control laws was assessed and the dislocated feedback control strategy was shown to be able to drive the system states toward desired setpoints quickly. Furthermore, the validity of the dislocated feedback control design was doubly ensured by the sliding mode control theory. It was found that those feedback schemes which manipulate cerium ion concentration can be practically realized by means of electrochemical oxidation or oxygen aeration. Our results were confirmed by the Simulink software package and the block diagram representations are included in the paper.

Purpose – The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method. Design/methodology/approach – The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem. Findings – The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat. Originality/value – The paper presents a new solution algorithm for the Stefan problem.

An efficient method based on the Faddeev-Leverrier algorithm combined with the Adomian decomposition method is devised to facilitate the stability analysis of multi-input multi-output control systems. In contrast to prior eigenvalue algorithms, our method affords all eigenvalues of the state matrix, either real or complex. Specifically, the calculation of the complex eigenvalues is made possible through special canonical forms, mainly involving square root operators, of the characteristic equation of the state matrix. Moreover, the proposed method does not require an initial guess, which is often a matter of concern since an inappropriate guess can cause failure in such available schemes. For the sake of illustration, a number of numerical examples, including chemical reaction processes, are also provided that demonstrate the efficiency of our new technique.

In this article, we extend the second–order Emden–Fowler equation to three new fourth–order Emden–Fowler type equations. We show that the singular point at x = 0, and hence the shape factor, may appear in more than one term of the equation. We solve the developed singular fourth–order equations by using the systematic Adomian decomposition method. We corroborate this work by studying several fourth–order Emden–Fowler type examples including linear and nonlinear cases.

In this paper, the flow of hydromagnetic non-Newtonian fluid under couple stresses through a porous channel is investigated using the Eyring-Powell model. The fluid is driven by an axial constant pressure gradient. Approximate solutions of the nonlinear dimensionless equations governing the fluid flow are obtained using a new modification of Adomian decomposition method (ADM). The effects of the variation of various flow parameters on both the velocity and temperature fields are deduced and discussed including surface-fluid interface friction and rate of heat transfer.

In this paper, we examine a system of two coupled nonlinear differential equations that relates the concentrations of carbon dioxide CO\(_2\) and phenyl glycidyl ether in solution. This system is subject to a set of Dirichlet boundary conditions and a mixed set of Neumann and Dirichlet boundary conditions. We apply the Adomian decomposition method combined with the Duan–Rach modified recursion scheme to analytically treat this system of coupled nonlinear boundary value problems. The rapid convergence of our analytic approximate solutions is demonstrated by graphs of the objective error analysis instead of comparison to an alternate solution technique alone. The Adomian decomposition method yields a rapidly convergent, easily computable, and readily verifiable sequence of analytic approximate solutions that is suitable for numerical parametric simulations. Thus our sequence of approximate solutions are shown to identically satisfy the original set of model equations as closely as we please.

The Hall-Yarborough equation (H-Y equation) of state has been favoured in natural gas engineering due to its accuracy and conciseness for many years. In this paper, the Adomian decomposition method (ADM) is employed to devise a novel algorithm for calculating the compressibility factors of natural gases through this reliable equation of state. A convergence accelerator technique, namely the efficient Shanks transform, is also exploited to further improve our scheme in terms of computational speed. Unlike most of the previous numerical solution strategies, our algorithm does not require an initial guess as the starting point and is computationally efficient. The proposed algorithm is found to be superior over the common Newton-Raphson algorithm, where we have also demonstrated that the latter can easily lead to grossly erroneous solutions. For the sake of illustration, a number of real-world case study problems are solved by our algorithm and relevant comparisons are provided.

In this study, an efficient iterative algorithm is devised to handle a nonlinear equation arising in estimation of thermodynamic properties at supercritical conditions. The approach is based on a synergistic combination of the classic Newton-Raphshon algorithm and the Adomian decomposition method. We demonstrate that the proposed method enjoys a higher degree of accuracy while requiring fewer iterations to reach a specific solution compared to that by the Newton-Raphson algorithm. To illustrate the efficiency of the aforementioned solution technique, several numerical examples are provided. The proposed method has been easily implemented in computer codes to provide parametric, not just numeric, solutions to the model equations. Consequently, one can derive other thermodynamic properties, which have not been treated parametrically to date, based on our new combined approach.

We investigate the reactant diffusion within a porous catalytic pellet including the heat of reaction. The Lane–Emden boundary value problem with an Arrhenius reaction rate is used to model the reactant concentration. We combine the Volterra integral form with the Adomian decomposition method to solve the equivalent Fredholm–Volterra integral equation. We then estimate the reactant concentration at the center of the catalytic pellet and the iterated Shanks transform is used to improve the accuracy of the approximations. The objective error analysis formulas are used to demonstrate a high accuracy and rapid rate of convergence, which does not depend on a priori knowledge of the exact solution or comparison with an alternate approximation method. Thus low-stage approximations by the Adomian decomposition method are validated for parametric simulations of the reactant concentration profiles and the effectiveness factor profiles. Our approach demonstrates enhancements over previous investigations, and is read...

The Casimir force can induce instability and adhesion in freestanding nanostructures. Previous research efforts in this area have exclusively focused on modeling the instability in structures with planar or rectangular cross-section, while, to the best knowledge of the authors, no attention has been paid to investigate this phenomenon for nanowires with circular cross-section. In this study, effects of the Casimir force on the instability and adhesion of freestanding Cylinder–Plate and Cylinder–Cylinder geometries are investigated, which are commonly encountered in real nanodevices. To compute the Casimir force, two approaches, i.e. the proximity force approximation (PFA) for small separations and Dirichlet asymptotic approximation (scattering theory) for large separations, are considered. A continuum mechanics theory is employed, in conjunction with the Euler-beam model, to obtain constitutive equations of the systems. The governing nonlinear constitutive equations of the nanostructures are solved using two different approaches, i.e. the analytical modified Adomian decomposition (MAD) and the numerical finite difference method (FDM). The detachment length and minimum gap, both of which prevent the Casimir force-induced adhesion, are computed for both configurations.

In this paper, we present new alternate derivations for the Volterra integral forms of the Lane-Emden equation that we derived in our last work (Wazwaz et al. in Appl Math Comput 219:5004–5019, 2013). The main focus will be on Lane-Emden equations for the shape factor of \(k=1\), where an alternate derivation for L’Hospital’s formula will be developed. Following our approach, the Adomian decomposition method, which provides an efficient algorithm for analytic approximate solutions of the Lane-Emden equation, will be used. Our results are supported by investigating several numerical examples that include linear and nonlinear initial value problems.

The use of the Hankinson–Thomas–Phillips correlation for prediction of the natural gas compressibility factor is a common practice in natural gas engineering calculations. However, this equation suffers a serious deficiency from a computational viewpoint; in that it is not explicit with respect to the z -factor and hence is subject to time-consuming trial and error procedures. In this paper, we propose an explicit series expansion equivalent to the Hankinson–Thomas–Phillips equation by the aid of a powerful mathematical technique known as the Adomian decomposition method. Furthermore, we have enhanced our formula by a applying nonlinear convergence accelerator algorithm, namely the Shanks transform. The proposed equation is simple, easy to use, and is shown to be extremely accurate in reproducing the experimental PVT data of natural gases. Moreover, in contrast to the previous numerical algorithms such as the Newton–Raphson algorithm, the explicit nature of our formula obviates the need for any initial guess as an input for calculation of the z -factor. Such independence permits our formula to always quickly converge to the correct z -factor.

In this paper, we present a new approach to solve nonlocal initial-boundary value problems for the Lotka-von Foerster model subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problem into an equivalent local initial-boundary value problem. Then we apply a modified Adomian decomposition method, which permits convenient resolution of these problems. Also a new method is developed to solve this nonlocal model, which reduces it to an equivalent local initial-boundary value problem for a system of ordinary differential equations.

In this paper, we present a new approach to solve nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local Dirichlet initial-boundary value problems, and then use a relatively new modified Adomian decomposition method (ADM). Furthermore we investigate the Fourier-Adomian method, which also does not require any a priori assumptions on the solution, for the solution of nonlocal initial-boundary value problems combined with our new approach. Several examples are presented to demonstrate the efficiency of the ADM.

In this paper, we use the systematic modified Adomian decomposition method (ADM) and the phenomenon of the self-canceling ”noise” terms for solving nonlinear weakly-singular Volterra, Fredholm, and Volterra-Fredholm integral equations. We show that the proposed approach minimizes the computation, when compared with other conventional schemes. Our results are validated by investigating several examples.

In this paper we propose a new modified recursion scheme for the resolution of boundary value problems (BVPs) for second-order nonlinear ordinary differential equations with Robin boundary conditions by the Adomian decomposition method (ADM). Our modified recursion scheme does not incorporate any undetermined coefficients. We also develop the multistage ADM for BVPs encompassing more general boundary conditions, including Neumann boundary conditions.

In this paper, the pull-in instability of a cantilever nano-actuator is considered incorporating the influence of surface effects, the fringing field and the Casimir attraction force. The instability parameters of the actuator are determined analytically under the assumption of a second-degree shape function for the beam during deflection. The influence of surface effects, the Casimir force and the fringing field effects on the pull-in parameters is investigated. The results demonstrate that the Casimir force decreases the pull-in deflection and voltage, the fringing field effects increase the pull-in deflection and decrease the pull-in voltage. The critical value of the surface effect parameter decreases monotonically from η ∗ = 4 as the Casimir force parameter increases. In the presence of the Casimir force, the surface effects decrease the pull-in deflection and voltage. For the MEMS model, which neglects the intermolecular forces, the surface effects do not influence the pull-in deflection, but decrease the pull-in voltage. For freestanding nanoactuators, the critical values of the tip deflection and the Casimir force parameter are obtained, and the surface effect parameter η decreases linearly with the critical value of the Casimir force parameter.

In this paper, we consider the coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. First, we utilize systems of Volterra integral forms of the Lane–Emden equations and derive the modified recursion scheme for the components of the decomposition series solutions. The numerical results display that the Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. The error analysis of the sequence of the analytic approximate solutions can be performed by using the error remainder functions and the maximal error remainder parameters, which demonstrate an approximate exponential rate of convergence.

In this paper, we present the Adomian decomposition method and its modifications combined with convergence acceleration techniques, such as the diagonal Pade approximants and the iterated Shanks transforms, to solve nonlinear fractional ordinary differential equations. Two nonlinear numeric examples demonstrate that either the diagonal Pade approximants or the iterated Shanks transforms can efficiently extend the effective convergence region of the decomposition series solution.

In this paper, the static response and pull-in instability of nanotweezers fabricated from carbon nanotubes (CNT) are theoretically investigated considering the effects of the Coulomb electrostatic and van der Waals molecular attractions. For this purpose, a nanoscale continuum model is employed to obtain the nonlinear constitutive equation of this nano-device. The van der Waals attraction is computed from the simplified Lennard-Jones potential. In order to solve the nonlinear constitutive equation of the nanotweezers, three different approaches, e.g. developing a lumped parameter model, applying the analytical modified Adomian decomposition (MAD) and using a commercial numerical integration routine, are employed. The obtained results are in good agreement with experimental measurements as reported in the literature. As a case study, we have investigated a freestanding nanotweezer and have determined the detachment length and minimum initial gap. Furthermore, range of dominancy of the molecular attraction has been discussed.

PurposeThe purpose of this paper is to present a new approach to solve nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.Design/methodology/approachThe authors first transform the given nonlocal boundary value problems of first‐ and second‐order differential equations into local boundary value problems of second‐ and third‐order differential equations, respectively. Then a modified Adomian decomposition method is applied, which permits convenient resolution of these equations.FindingsThe new technique, as presented in this paper in extending the applicability of the Adomian decomposition method, has been shown to be very efficient for solving nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.Originality/valueThe paper presents a new solution algorithm for the nonlocal boundary value problems of linear and nonlinear first‐ and second‐order differential equations subject to nonlocal conditions of integral type.

In this paper, we introduce systems of Volterra integral forms of the Lane–Emden equations. We use the systematic Adomian decomposition method to handle these systems of integral forms. The Volterra integral forms overcome the singular behavior at the origin x = 0. The Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. Our results are supported by investigating several numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper we solve the common nonlinear boundary value problems (BVPs) of cantilever-type micro-electromechanical system (MEMS) and nano-electromechanical system (NEMS) using the distributed parameter model by the Duan–Rach modified Adomian decomposition method (ADM). The nonlinear BVPs that are investigated include the cases of the single and double cantilever-type geometries under the influence of the intermolecular van der Waals force and the quantum Casimir force for appropriate distances of separation. The new Duan–Rach modified ADM transforms the nonlinear BVP consisting of a nonlinear differential equation subject to appropriate boundary conditions into an equivalent nonlinear Fredholm–Volterra integral equation before designing an efficient recursion scheme to compute approximate analytic solutions without resort to any undetermined coefficients. The new approach facilitates parametric analyses for such designs and the pull-in parameters can be estimated by combining with the Pade approximant. We also consider the accuracy and the rate of convergence for the solution approximants of the resulting Adomian decomposition series, which demonstrates an approximate exponential rate of convergence. Furthermore we show how to easily achieve an accelerated rate of convergence in the sequence of the Adomian approximate solutions by applying Duan's parametrized recursion scheme in computing the solution components. Finally we compare the Duan–Rach modified recursion scheme in the ADM with the method of undetermined coefficients in the ADM for solution of nonlinear BVPs to illustrate the advantages of our new approach over prior art.