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The purpose of this paper is to consider divergence of composite plate wings as well as slender wings with thin-walled cross-section of small-size airplanes. The main attention is paid to establishing of closed-form mathematical solutions for models of wings with coupling effects. Simplified solutions for calculating the divergence speed of wings with different geometry are established.

The wings are modeled as anisotropic plate elements and thin-walled beams with closed cross-section. Two-dimensional plate-like models are applied to analysis and design problems for wings of large aspect ratio.

At first, the equations of elastic deformation for anisotropic slender, plate-like wing with the large aspect ratio are studied. The principal consideration is delivered to the coupled torsion-bending effects. The influence of anisotropic tailoring on the critical divergence speed of the wing is examined in closed form. At second, the method is extended to study the behavior of the large aspect ratio, anisotropic wing with box-like wings. The static equations of the wing with box-like profile are derived using the theory of anisotropic thin-walled beams with closed cross-section. The solutions for forward-swept wing with box-like profiles are given in analytical formulas. The formulas for critical divergence speed demonstrate the dependency upon cross-sectional shape characteristics and anisotropic properties of the wing.

The following simplifications are used: the simplified aerodynamic theory for the wings of large aspect ratio was applied; the static aeroelastic instability is considered (divergence); according to standard component methodology, only the component of wing was modeled, but not the whole aircraft; the simplified theories (plate-lime model for flat section or thin-walled beam of closed-section) were applied; and a single parameter that defines the rotation of a stack of single layers over the face of the wing.

The simple, closed-form formulas for an estimation of critical static divergence are derived. The formulas are intended for use in designing of sport aircraft, gliders and small unmanned aircraft (drones). No complex analysis of airflow and advanced structural and aerodynamic models is necessary. The expression for chord length over the span of the wing allows for accounting a board class of wing shapes.

The derived theory facilitates the use of composite materials for popular small-size aircraft, and particularly, for drones and gliders.

The closed-form solutions for thin-walled beams in steady gas flow are delivered in closed form. The explicit formulas for slender wings with variable chord and stiffness along the wing span are derived.

The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. Combination of the Laplace transform and homotopy perturbation methods (LTHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

The authors present the solution of nonlinear Boussinesq equation by combination of Laplace transform and new homotopy perturbation methods. An important property of the proposed method, which is clearly demonstrated in example, is that spectral accuracy is accessible in solving specific nonlinear nonlinear Boussinesq equation which has analytic solution functions.

The authors proposed a combination of Laplace transform method and homotopy perturbation method to solve the one-dimensional Boussinesq equation. The results are found to be in excellent agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences.

The authors provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. In this work combination of Laplace transform and new homotopy perturbation methods (LTNHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

This paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).

The proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.

To demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.

The H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

Fibre‐reinforced plastic (FRP) materials have been recognised as new innovative materials for concrete rehabilitation and retrofit. Since concrete is poor in tension, a beam without any form of reinforcement will fail when subjected to a relatively small tensile load. Therefore, the bonding of FRP plate to reinforced concrete (RC) structure is an effective solution to increase its overall strength. In such plated beams, tensile forces develop in the bonded plate and these have to be transferred to the original beam via interfacial shear and normal stresses. Consequently, the debonding of FRP plates bonded to reinforced concrete beams is believed to be initiated by the stress concentration in the adhesive layer. Accurate predictions of the interfacial stresses are prerequisite for designing against debonding failures. In the present analysis, a simple theoretical model to estimate shear and normal stresses is proposed, including the variation in FRP plate fibre orientation. The solution shows significant shear and normal stresses concentration at the plates end. A parametrical study is carried out to show the effects of some design variables, e.g., thickness of adhesive layer and FRP plate, and the distance from support to cut ‐ off end of bonded plates.

In this paper, we develop a model for machine maintenance inspection problem. Conditions under which an economical solution exists are derived. An optimal solution procedure yielding the optimal inspection interval length in few iterations is proposed as well as a closed form heuristic solution which provides a very good approximation. An example is used to illustrate both approaches.

The objective of this paper is to construct a continuous model for the thermo‐visco‐elastic contact of a nominal flat, non‐smooth, punch and a smooth surface of a rigid half‐space. The considered model aims at studying the normal approach as a function of the applied loads and temperatures. The proposed model assumes the punch surface material to behave according to the linear Kelvin‐Voigt visco‐elastic material. The punch surface, which is known to be fractal in nature, is modeled in this work using a deterministic Cantor structure. An asymptotic power low, deduced using approximate iterative relations, is used to express the punch surface approach as a function of the remote forces and bulk temperatures when the approach of the punch surface and the half space is in the order of the size of the surface roughness. The results obtained using this model, which admits closed form solution, are displayed graphically for selected values of the system parameters; the fractal surface roughness and various material properties. The obtained results showed good agreement with published experimental results.

The purpose of this paper is to present a numerical and an analytical study of the fluid flow and heat transfer in the unsteady, laminar boundary layer resulting from the forced convection flow along a semi‐infinite wedge, where the transients are initiated at time t¯ = 0 when the wedge is impulsively started from rest with a uniform velocity and a constant heat flux at the walls of the wedge is suddenly imposed.

The velocity of the main free stream is written in non‐dimensional form for t > 0 as u_e(x) = x^m, where x is the non‐dimensional distance along the surface from the leading edge (apex) of the wedge and the constant m is related to the included angle of the wedge πβ by m = β / (2 − β) (0 ≤ m ≤ 1 for physical applications). The wedge and the fluid are assumed to be initially (t¯ = 0) at the same uniform temperature T_∞, so that there is zero heat flux at the surface. A time‐dependent thermal boundary layer is then produced at t¯ = 0 as the zero heat flux at the surface is suddenly changed, and a constant heat flux q_w is imposed as the wedge is set into motion. Analytical solutions for the simultaneous development of the momentum and thermal boundary layers are obtained for both small (initial unsteady flow) and large (steady‐state flow) times for several wedge angles (several values of m) and several values of the Prandtl number Pr. These two asymptotic solutions are matched using two specialised numerical procedures.

The numerical results obtained for the transient fluid velocity and temperature fields concentrate mainly on the case when the Prandtl number Pr = 1 and m = 1 / 5, namely a wedge angle of 60^○. Required alterations to these parameters are then discussed with reference to variations in Pr and m separately. Further, an engineering empirical expression is presented for the skin friction C_f (τ) Re_x^1/2 that is valid for all times. The comparison between the empirical formula and the full numerical solution demonstrates that this matching solution can be used with confidence over the whole range of values of the non‐dimensional time τ for each of the values of m presented, and may therefore be used with confidence in engineering applications.

The results of the present work, which have been obtained through many computations, are very important for the advancement of knowledge on this classical problem of fluid mechanics and heat transfer. It is believed that such very detailed solutions have not previously been presented.

Interstellar gas passing through the solar system may effect the interplanetary gas, planetary atmospheres and satellite orbits. Interaction of the interstellar and interplanetary gases is considered; a solar system corona may be formed.

The purpose of this paper is to develop a new improved solution and a new model to predict both shear and normal interfacial stress in simply supported beams strengthened with bonded prestressed FRP laminates by taking into account the fiber volume fraction spacing that play an important role on the interfacial stresses concentration.

The study has been conducted by using analytical approaches for interfacial stresses in plated beams. The analysis is based on the deformation compatibility approach where both the shear and normal stresses are assumed to be invariant across the adhesive layer thickness. In addition, an unrealistic restriction of the same curvatures in the RC beam and FRP panel commonly used in most of the existing studies is released in the present theoretical formulation.

To verify the analytical model, the present predictions are compared first with those of (Malek et al., 1998; Smith and Teng, 2001) in the case of the absence of the prestressing force; for the second time, the present method is compared with that developed by (Al-Emrani and Kliger, 2006; Benachour et al., 2008) in the case where only the prestressing force is applied. From the presented results, it can be seen that the present solution agree closely with the other methods in the literature.

The paper puts in evidence a new originality approach theory, taking into account the mechanical load, and the prestressed FRP plate model having variable fiber spacing which considers a strength rigidity and resistance of the damaged structures, which is one aspect that has not been taken into account by the previous studies.