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The steady-state periodic flow of Stokes’ second problem for viscoelastic fluids with constitutive equation in terms of the distributed-order derivative was considered. The distributed-order derivative involves an integration with respect to the order of fractional derivative, and the order is associated with a weight function p ( α ). With a general weight function p ( α ), the flow velocity was obtained. The amplitude, the penetration depth and the wavelength were given analytically. Results of Newtonian fluid and single fractional constitutive equation were derived as special cases of weight function p ( α ). Also we considered other three cases of weight function p ( α ): linear combination of Dirac delta functions, constant and parameterized exponential function.

The steady-state response to periodic excitation in the linear fractional vibration system was considered by using the fractional derivative operator . First we investigated the response to the harmonic excitation in the form of complex exponential function. The amplitude-frequency relation and phase-frequency relation were derived. The effect of the fractional derivative term on the stiffness and damping was discussed. For the case of periodic excitation, we decompose the periodic excitation into a superposition of harmonic excitations by using the Fourier series, and then utilize the results for harmonic excitations and the principle of superposition, where our adopted tactics avoid appearing a fractional power of negative numbers to overcome the difficulty in fractional case. Finally we demonstrate the proposed method by three numerical examples.

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.

Recently, the reproducing kernel Hilbert space methods(RKHSM) (see Wang et al (2011) 2; Lin and Lin (2010) 3; Wu and Li (2011) 4; Zhou et?al. (2009) 6; Jiang and Chen (2014) 7; Wang et?al. (2010) 8; Du and Cui (2008) 9; Akram et?al. (2013) 10; Lu and Cui (2010) 11; Wang et al. (2008) 12; Yao and Lin (2009) 13; Geng et?al. (2014) 14 ; Arqub et?al. (2013) 15) emerged one after the other. But, a lot of difficult work should be done to deal with multi-point boundary value problems(BVPs). Our work is aimed at giving a new reproducing kernel method for multi-point BVPs. We do not put the homogenization conditions into the reproducing kernel space which can avoid to compute the reproducing kernel satisfying boundary conditions and the orthogonal system. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate that new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

Stokes’ second problem is about the steady-state oscillatory flow of a viscous fluid due to an oscillating plate. We consider Stokes’ second problem for a class of viscoelastic fluids that are characterized by a fractional constitutive equation. The exact analytical solution as parametrized by the order of the fractional derivative is obtained. We provide detailed analyses and discussions for effects of the model parameters on the wave length and the amplitude in the flow field. We show that, as the order varies from 0 to 1, the flow displays a transition from elastic to viscous behavior. Finally, we consider the case of the constitutive equation for a fractional element or a spring-pot in series with a dashpot.

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.

Based on Adomian decomposition method, a new algorithm for solving boundary value problem (BVP) of nonlinear partial differential equations on the rectangular area is proposed. The solutions obtained by the method precisely satisfy all boundary conditions, except the small pieces near the four corners of the rectangular area. A theorem on the boundary error is given. Hence, the Adomian decomposition method is more efficiently applied to BVPs for partial differential equations. Segmented and weighted analytical solutions with a high accuracy for the BVP of nonlinear groundwater equations on a rectangular area are obtained by the present algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

Exact solutions of shear stress versus velocity gradient and the numerical solutions of streamwise velocity distribution in radial direction of a Johnson–Segalman fluid in a circular pipe are obtained. The effects of material parameters, Weissenberg number, ratio of viscosities and slip parameter, on shear stress and streamwise velocity have been considered to investigate the discontinuous velocity derivatives and stick-slip phenomenon at the wall. We find that there is a non-monotonic relationship between the shear stress and rate of shear for certain values of the material parameters and consequently, the velocity profile has discontinuous derivatives. Also the non-monotonic region is shortening with the increasing of ratio of viscosities, and is lengthening with the increasing of slip parameter. For the flow with larger slip parameter, the initially driving pressure gradient has to be larger for the appearance of spurt phenomenon. Moreover, the variational range of material parameters is given for the appearance of a non-monotonic relationship between the shear stress and the rate of shear. Finally, we have shown the exact expression of critical pressure gradient and also have given the conditions where spurt phenomena occur.

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 nonlinear differential equation for temperature distribution of convective straight fins with temperature-dependent thermal conductivity is solved by using a new modified decomposition method (MDM) for boundary value problems. In the new MDM the recursion scheme of the solution components does not involve any undetermined coefficients. Using the new method, the temperature distribution and the efficiency of the fin can be expressed analytically as functions containing two fin parameters without any undetermined coefficients, which greatly facilitates parameter analysis.

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.

We present a new approach to calculate analytic approximations of blow-up solutions and their critical blow-up times. Our approach applies the Adomian decomposition–Pade method to quickly and easily compute the critical blow-up times, which comprises the Adomian decomposition method combined with the Pade approximants technique. We validate our new approach with a variety of numerical examples, including nonlinear ODEs, systems of nonlinear ODEs, and nonlinear PDEs. Furthermore, our new method is shown to be more convenient than prior art that relies on compound discretized algorithms. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we use the systematic Adomian decomposition method to handle the integral form of the Lane-Emden equations with initial values and boundary conditions. The Volterra integral form of the Lane-Emden equation overcomes the singular behavior at the origin x=0. We confirm our belief that the Adomian decomposition method provides efficient algorithm for analytic approximate solutions of the equation. Our results are supported by investigating several numerical examples that include initial value problems and boundary value problems as well. Finally we consider the modified decomposition method of Rach, Adomian and Meyers for the Volterra integral form.

In this paper, we develop new numeric modified Adomian decomposition algorithms by using the Wazwaz–El-Sayed modified decomposition recursion scheme, and investigate their practicality and efficiency for several nonlinear examples. We show how we can conveniently generate higher-order numeric algorithms at will by this new approach, including, by using examples, 12th-order and 20th-order numeric algorithms. Furthermore, we show how we can achieve a much larger effective region of convergence using these new discrete solutions. We also demonstrate the superior robustness of these numeric modified decomposition algorithms including a 4th-order numeric modified decomposition algorithm over the classic 4th-order Runge–Kutta algorithm by example. The efficiency of our subroutines is guaranteed by the inclusion of the fast algorithms and subroutines as published by Duan for generation of the Adomian polynomials to high orders.

In this paper we present the generalized Adomian–Rach theorem and the generalized Rach–Adomian–Meyers modified decomposition method for solving multi-order nonlinear fractional ordinary differential equations. We consider different classes of initial value problems for nonlinear fractional ordinary differential equations, including the case of real-valued orders and another case of rational-valued orders, which are solved by the present method. This method can treat any analytic nonlinearity. The coefficients of the solution in the form of a generalized power series are determined by a convenient recurrence scheme, which does not involve integration operations compared with the classic Adomian decomposition method.

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.

In this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition. By analyzing and transforming the Taylor expansion of the nonlinearity about the initial solution component, the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of Diophantine equations. This conversion simplifies the discussion and presents a new idea for decompositions. We enumerate five types of partitions and their corresponding decomposition polynomials. Each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums. In Types 2, 3 and 4, there is a parameter q and each value of q corresponds to a class of decomposition polynomials. In Type 5, each positive integer sequence { c j } satisfying 1 = c 1 ⩽ c 2 ⩽ ⋯ and j ⩽ c j for j = 2, 3, … corresponds to a class of decomposition polynomials. Four classes of the Adomian polynomials [R. Rach, A new definition of the Adomian polynomials, Kybernetes 37 (2008) 910–955] are derived as particular cases.

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach–Adomian–Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge–Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Pade approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge–Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge–Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge–Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.

In this article, we present new algorithms for the nonclassic Adomian polynomials, which are valuable for solving a wide range of nonlinear functional equations by the Adomian decomposition method, and introduce their symbolic implementation in MATHEMATICA. Beginning with Rach's new definition of the Adomian polynomials, we derive the explicit expression for each class of the Adomian polynomials, e.g. A"[email protected]?"k"="1^mf^(^k^)(u"0)Z"m","k for the Class II, III and IV Adomian polynomials, where the Z"m","k are called the reduced polynomials. These expressions provide a basis for developing improved algorithmic approaches. By introducing the index vectors, the recurrence algorithms for the reduced polynomials are suitably deduced, which naturally lead to new recurrence algorithms for the Class II and Class III Adomian polynomials. MATHEMATICA programs generating these classes of Adomian polynomials are subsequently presented. Computation shows that for computer generation of the Class III Adomian polynomials, the new algorithm reduces the running times compared with the definitional formula. We also consider the number of summands of these classes of Adomian polynomials and obtain the corresponding formulas. Finally, we demonstrate the versatility of the four classes of Adomian polynomials with several examples, which include the nonlinearity of the form f(t,u), explicitly depending on the argument t.

Inclines are additively idempotent semirings, in which the partial order ≤ : x ≤ y if and only if x + y = y is defined and products are less than or equal to either factor. Boolean algebra, max-min fuzzy algebra, and distributive lattices are examples of inclines. In this article, standard bases of a finitely generated vector space over a linearly ordered commutative incline are studied. We obtain that if a standard basis exists, then it is unique. In particular, if the incline is solvable or multiplicatively-declined or multiplicatively-idempotent (i.e., a chain semiring), further results are obtained, respectively. For a chain semiring a checkable condition for distinguishing if a basis is standard is given. Based on the condition an algorithm for computing the standard basis is described.

In this article we present four analytic recurrence algorithms for the multivariable Adomian polynomials. As special cases, we deduce the four simplified results for the one-variable Adomian polynomials. These algorithms are comprised of simple, orderly and analytic recurrence formulas, which do not require time-intensive operations such as expanding, regrouping, parametrization, and so on. They are straightforward to implement in any symbolic software, and are shown to be very efficient by our verification using MATHEMATICA 7.0. We emphasize that from the summation expressions, A n = ∑ k = 1 n U n k for the multivariable Adomian polynomials and An = ∑ k = 1 n f ( k ) ( u 0 ) C n k for the one-variable Adomian polynomials, we obtain the recurrence formulas for the U n k and the C n k . These provide a theoretical basis for developing new algorithmic approaches such as for parallel computing. In particular, the recurrence process of one particular algorithm for the one-variable Adomian polynomials does not involve the differentiation operation, but significantly only the arithmetic operations of multiplication and addition are involved; precisely C n 1 = u n ( n ⩾ 1 ) and C n k = 1 n ∑ j = 0 n - k ( j + 1 ) u j + 1 C n - 1 - j k - 1 ( 2 ⩽ k ⩽ n ) . We also discuss several other algorithms previously reported in the literature, including the Adomian–Rach recurrence algorithm [1] and this author’s index recurrence algorithm [23] , [36] .

In this article the sum of the series of multivariable Adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable Taylor expansion of an analytic function of the decomposition series of solutions u 1 , u 2 , … , u m about the initial solution components u 1,0 , u 2,0 , … , u m ,0 ; of course the multivariable Adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations. The index matrices and their simplified forms of the multivariable Adomian polynomials are introduced. We obtain the recurrence relations for the simplified index matrices, which provide a convenient algorithm for rapid generation of the multivariable Adomian polynomials. Another alternative algorithm for term recurrence is established. In these algorithms recurrence processes do not require complicated operations such as parametrization, expanding and regrouping, derivatives, etc. as practiced in prior art. The MATHEMATICA program generating the Adomian polynomials based on the algorithm in this article is designed.

In this paper a recurrence technique for calculating Adomian polynomials is proposed, the convergence of the series for the Adomian polynomials is discussed, and the dependence of the convergent domain of the solution's decomposition series @?"n"="0^~u"n on the initial component function u"0 is illustrated. By introducing the index vectors of the Adomian polynomials the recurrence relations of the index vectors are discovered and the recurrence triangle is given. The method simplifies the computation of the Adomian polynomials. In order to obtain a solution's decomposition series with larger domain of convergence, we illustrate by examples that the domain of convergence can be changed by choosing a different u"0 and a modified iteration.

Generalized fuzzy matrices are considered as matrices over a special type of semiring which is called an incline, and their transitive closure, powers, determinant and adjoint matrices are studied. An expression for the transitive closure of a matrix A as a sum of its powers and some su2cient conditions for powers of a matrix to converge are given. If the incline is commutative, a su2cient condition for nilpotency of a matrix is obtained, namely the determinants of the principal submatrices of the matrix are all equal to zero element. In addition, it is proved that A n−1 is equal to the adjoint matrix of A if the matrix A satis5es A ? In.

The fundamental solution for time- and space-fractional partial differential operator Dtλ+a2(−▵)γ∕2(λ,γ>0) is given in terms of the Fox’s H-function. Here the time-fractional derivative in the sense of generalized functions (distributions) Dtλ is defined by the convolution Dtλf(t)=Φ−λ(t)*f(t), where Φλ(t)=t+λ−1∕Γ(λ) and f(t)≡0 as t