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Article
Publication date: 19 June 2020

## Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

### Abstract

#### Purpose

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

#### Design/methodology/approach

The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.

#### Findings

When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.

#### Research limitations/implications

Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.

#### Practical implications

Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.

#### Originality/value

Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

### Details

Engineering Computations, vol. 38 no. 1
Type: Research Article
ISSN: 0264-4401

### Keywords

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Article
Publication date: 23 November 2018

## Approximate static aeroelastic analysis of composite wings

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…

### Abstract

#### Purpose

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.

#### Design/methodology/approach

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.

#### Findings

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.

#### Research limitations/implications

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.

#### Practical implications

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.

#### Social implications

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

#### Originality/value

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.

### Details

Multidiscipline Modeling in Materials and Structures, vol. 15 no. 2
Type: Research Article
ISSN: 1573-6105

### Keywords

Content available
Article
Publication date: 15 December 2020

## Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics

Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions…

Open Access

### Abstract

#### Purpose

Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.

#### Design/methodology/approach

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=i=0nai(DξαG/G)i/i=0nbi(DξαG/G)i.

#### Findings

Achieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.

#### Originality/value

The rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

### Details

Arab Journal of Mathematical Sciences, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 1319-5166

### Keywords

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Article
Publication date: 8 August 2016

## Lattice of infinite bending-resistant fibers

The purpose of this paper is to introduce the double-periodic lattice, composed of bending-resistant fibers. The essence of the model is that the filaments are of infinite…

### Abstract

#### Purpose

The purpose of this paper is to introduce the double-periodic lattice, composed of bending-resistant fibers. The essence of the model is that the filaments are of infinite length and withstand tension and bending. The constitutive equations of the lattice in discrete and differential formulations are derived. Two complementary systems of loads, which cause different deformation two orthogonal families of fibers, occur in the lattice. The fracture behavior of the material containing a semi-infinite crack is investigated. The crack problem reduces to the exactly solvable Riemann-Hilbert problem. The solution demonstrates that the behavior of material cardinally depends upon the tension in the orthogonal family of fibers. If tension in fibers exists, opening of the crack under action of loads in two-dimensional lattice is similar to those in elastic solid. In the absence of tension, contrarily, there is a finite angle between edges at the crack tip.

#### Design/methodology/approach

The description of stress state in the crack vicinity is reduced to the solution of mixed boundary value problem for simultaneous difference equations. In terms of Fourier images for unknown functions the problem is equivalent to a certain Riemann-Hilbert problem.

#### Findings

The analytical solution of the problem shows that fracture behavior of the material depends upon the presence of stabilizing tension in fibers, parallel to crack direction. In the presence of tension in parallel fibers fracture character of two-dimensional lattice is similar to behavior of elastic solid. In this case the condition of crack grows can be formulated in terms of critical stress intensity factor. Otherwise, in the absence of stabilizing tension, the crack surfaces form a finite angle at the tip.

#### Research limitations/implications

Linear behavior of fibers until rupture. Small deflections. Perfect two-dimensional lattice.

#### Practical implications

The model provides exact analytical estimation of stresses on the crack tip as the function of fibers’ stiffness.

#### Originality/value

The model is the extension of known lattice models, taking into account the semi-infinite crack in the lattice. This is the first known closed form solution for an infinite lattice model with the crack.

### Details

Multidiscipline Modeling in Materials and Structures, vol. 12 no. 2
Type: Research Article
ISSN: 1573-6105

### Keywords

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Article
Publication date: 23 January 2009

## Aeroelasticity consideration of supersonic vehicle using closed form analytical aerodynamic model

The purpose of this paper is to investigate the aeroelastic behavior of a supersonic flight vehicle flying at moderate angles of attack using global analytic nonlinear…

### Abstract

#### Purpose

The purpose of this paper is to investigate the aeroelastic behavior of a supersonic flight vehicle flying at moderate angles of attack using global analytic nonlinear aerodynamic model.

#### Design/methodology/approach

Aeroelastic behavior of a supersonic flight vehicle flying at moderate angles of attack is considered, using nonlinear aerodynamics and linear elastodynamics and structural models. Normal force distribution coefficient over the length of the vehicle and pitching moment coefficient are the main aerodynamic parameters used in the aeroelastic modeling. It is very important to have closed form analytical relations for these coefficients in the model. They are generated using global nonlinear multivariate orthogonal modeling functions in this work. Angle of attack and length of the vehicle are selected as independent variables in the first step. Local variation of angle of attack is applied to the analytical model and due to its variation along the body of the vehicle, equations of motion are finalized. Mach number is added to the independent variables to investigate its role on instability of the vehicle and the modified model is compared with the previous one in the next step. Thrust effect on the aeroelastic stability of the vehicle is analyzed at final stage.

#### Findings

It is shown that for the vehicles having simple configurations and low length to diameter ratios flying at low angles of attack, assuming normal force distribution coefficient linear relative to α is reasonable. It is concluded that vehicle's velocity and thrust has not great effect on its divergence dynamic pressure.

#### Originality/value

Based on the constructed model, a simulation code is generated to investigate the aeroelastic behavior of the vehicle. The resultant code is verified by investigating the static aeroelastic stability margin of the vehicle presented in the references. Mach number effect on the aeroelastic behavior of the vehicle is considered using modified aerodynamic model and is compared with the results. Data base for identifying aerodynamic coefficients is conducted using CFD code.

### Details

Aircraft Engineering and Aerospace Technology, vol. 81 no. 2
Type: Research Article
ISSN: 0002-2667

### Keywords

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Article
Publication date: 1 June 2000

## Field methods and computational techniques

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic…

### Abstract

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

### Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 19 no. 2
Type: Research Article
ISSN: 0332-1649

### Keywords

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Article
Publication date: 20 November 2017

## Bond valuation for generalized Langevin processes with integrated Lévy noise

Recently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation…

### Abstract

#### Purpose

Recently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation (GLE). In this paper, the authors extend the model to a wider class of memory kernels and then propose a bond and bond option valuation model based on the extension of the generalized Langevin process of Stein et al. (2016).

#### Design/methodology/approach

Bond and bond option pricing based on the proposed interest rate models presents new difficulties as the standard partial differential equation method of stochastic calculus for bond pricing cannot be used directly. The authors obtain bond and bond option prices by finding the closed form expression of the conditional characteristic function of the integrated short rate process driven by a general Lévy noise.

#### Findings

The authors obtain zero-coupon default-free bond and bond option prices for short rate models driven by a variety of Lévy processes, which include Vasicek model and the short rate model obtained by solving a second-order Langevin stochastic differential equation (SDE) as special cases.

#### Originality/value

Bond and bond option pricing plays an important role in capital markets and risk management. In this paper, the authors derive closed form expressions for bond and bond option prices for a wider class of interest rate models including second-order SDE models. Closed form expressions may be especially instrumental in facilitating parameter estimation in these models.

### Details

The Journal of Risk Finance, vol. 18 no. 5
Type: Research Article
ISSN: 1526-5943

### Keywords

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Article
Publication date: 7 August 2019

## Assembly variation analysis of compliant mechanisms by combining direct linearization method and Lagrange’s equations

This paper aims to carry out assembly variation analysis for mechanisms with compliant joints by considering deformations induced by manufactured deviations. Such an…

### Abstract

#### Purpose

This paper aims to carry out assembly variation analysis for mechanisms with compliant joints by considering deformations induced by manufactured deviations. Such an analysis procedure extends the application area of direct linearization method (DLM) to compliant mechanisms and also illustrates the dimensional interaction within multi-loop compliant structures.

#### Design/methodology/approach

By applying DLM to both geometrical equations and Lagrange’s equations of the second kind, an analytical deviation modeling method for mechanisms with compliant joints are proposed and further used for statistical assembly variation analysis. The precision of this method is verified by comparing it with finite element simulation and traditional DLM.

#### Findings

A new modeling method is proposed to represent kinematic relationships between joint deformations and parts/components deviations. Based on a case evaluation, the computational efficiency is improved greatly while the modeling accuracy is maintained at more than 94% rate comparing with the benchmark finite element simulation.

#### Originality/value

The Equilibrium Equations of Incremental Forces derived from Lagrange’s equations are proposed to quantitatively represent the relationships between manufactured deviations and assembly deformations. The present method extends the application area of DLM to compliant structures, such as automobile suspension systems and some Micro-Electro-Mechanical-Systems.

### Details

Assembly Automation, vol. 39 no. 4
Type: Research Article
ISSN: 0144-5154

### Keywords

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Article
Publication date: 1 March 1992

## A FINITE ELEMENT EIGENVALUE METHOD FOR SOLVING TRANSIENT HEAT CONDUCTION PROBLEMS

An eigenvalue method is presented for solving the transient heat conduction problem with time‐dependent or time‐independent boundary conditions. The spatial domain is…

### Abstract

An eigenvalue method is presented for solving the transient heat conduction problem with time‐dependent or time‐independent boundary conditions. The spatial domain is divided into finite elements and at each finite element node, a closed‐form expression for the temperature as a function of time can be obtained. Three test problems which have exact solutions were solved in order to examine the merits of the eigenvalue method. It was found that this method yields accurate results even with a coarse mesh. It provides exact solution in the time domain and therefore has none of the time‐step restrictions of the conventional numerical techniques. The temperature field at any given time can be obtained directly from the initial condition and no time‐marching is necessary. For problems where the steady‐state solution is known, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed. These features lead to great savings in computation time, especially for problems with long time duration. Furthermore, the availability of the closed form expressions for the temperature field makes the present method very attractive for coupled problems such as solid—fluid and thermal—structure interactions.

### Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 2 no. 3
Type: Research Article
ISSN: 0961-5539

### Keywords

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Book part
Publication date: 1 January 2008

## Bayesian analysis of the consumption CAPM

In this paper we take up Bayesian inference for the consumption capital asset pricing model. The model has several econometric complications. First, it implies exact…

### Abstract

In this paper we take up Bayesian inference for the consumption capital asset pricing model. The model has several econometric complications. First, it implies exact relationships between asset returns and the endowment growth rate that will be rejected by all possible realizations. Second, it was thought before that it is not possible to express asset returns in closed form. We show that Labadie's (1989) solution procedure can be applied to obtain asset returns in closed form and, therefore, it is possible to give an econometric interpretation in terms of traditional measurement error models. We apply the Bayesian inference procedures to the Mehra and Prescott (1985) dataset, we provide posterior distributions of structural parameters and posterior predictive asset return distributions, and we use these distributions to assess the existence of asset returns puzzles. The approach developed here, can be used in sampling theory and Bayesian frameworks alike. In fact, in a sampling-theory context, maximum likelihood can be used in a straightforward manner.

### Details

Bayesian Econometrics
Type: Book
ISBN: 978-1-84855-308-8

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