• Journal of applied mathematics & informatics
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• v.7 no.2
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• pp.453-465
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• 2000

In this paper, a new method for finding the approximate solution of a second order nonlinear partial differential equation is introduced. In this method the problem is transformed to an equivalent optimization problem. them , by considering it as a distributed parameter control system the theory of measure is used for obtaining the approximate solution of the original problem.

• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1123-1139
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• 2014

In this work, we investigate the existence of the extremal solutions for a class of fractional partial differential equations with order 1 < ${\alpha}$ < 2 by upper and lower solution method. Using the theory of Hausdorff measure of noncompactness, a series of results about the solutions to such differential equations is obtained.

• Coupled systems mechanics
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• v.2 no.3
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• pp.255-269
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• 2013

This paper targets to investigate the solution of fuzzy quadratic Riccati differential equations with various types of fuzzy environment using Homotopy Perturbation Method (HPM). Fuzzy convex normalized sets are used for the fuzzy parameter and variables. Obtained results are depicted in term of plots to show the efficiency of the proposed method.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.14 no.2
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• pp.91-97
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• 2014

Chaotic dynamics is an active research area in fields such as biology, physics, sociology, psychology, physiology, and engineering. Interest in chaos is also expanding to the social sciences, such as politics, economics, and societal events prediction. Most people pursue happiness, both spiritual and physical in many cases. However, happiness is not easy to define, because people differ in how they perceive it. Happiness can exist in mind and body. Therefore, we need to be happy in both simultaneously to achieve optimal happiness. To do this, we need to synchronize mind and body. In this paper, we propose a chaotic synchronization method in a mathematical model of happiness organized by a second-order ordinary differential equation with external force. This proposed mathematical happiness equation is similar to Duffing's equation, because it is derived from that equation. We introduce synchronization method from our mathematical happiness model by using the derived Duffing equation. To achieve chaotic synchronization between the human mind and body, we apply an idea of mind/body unity originating in Oriental philosophy. Of many chaotic synchronization methods, we use only coupled synchronization, because this method is closest to representing mind/body unity. Typically, coupled synchronization can be applied only to non-autonomous systems, such as a modified Duffing system. We represent the result of synchronization using a differential time series mind/body model.

• Geomechanics and Engineering
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• v.30 no.4
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• pp.363-372
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• 2022

One of the most frequent issues in tunnel excavation is the collapse of rock blocks and the dropping of rock fragments from the tunnel face. The tunnel face can be reinforced using a number of techniques. One of the most popular and affordable solutions is the use of face longitudinal dowels, which has benefits including high strength, flexibility, and ease of cutting. In order to examine the reinforced face, this work shows the longitudinal deformation profile and ground response curve for a tunnel face. This approach is based on assumptions made during the analysis phase of problem solving. By knowing the tunnel face response and dowel behavior, the interaction of two elements can be solved. The rock element equation derived from the rock bolt method is combined with the dowel differential equation to solve the reinforced ground response curve (GRC). With a straightforward and accurate analytical equation, the new differential equation produces the reinforced displacement of the tunnel face at each stage of excavation. With simple equations and a less involved computational process, this approach offers quick and accurate solutions. The FLAC3D simulation has been compared with the suggested analytical approach. A logical error is apparent from the discrepancies between the two solutions. Each component of the equation's effect has also been described.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.23 no.8
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• pp.734-741
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• 2013

In contrast of external excitations, parametric excitations can produce a large response when the excitation frequency is away from the linear natural frequencies. The Mathieu equation is the simplest differential equation with periodic coefficients, which lead to the parametric excitation. The Mathieu equation may have the unbounded solutions. This work conducted the stability analysis for the Mathieu equation, using Floquet theory and numerical method. Using Lindstedt's perturbation method, harmonic solutions of the Mathieu equation and transition curves separating stable from unstable motions were obtained. Using Floquet theory with numerical method, stable and unstable regions were calculated. The numerical method had the same transition curves as the perturbation method. Increased stable regions due to the inclusion of damping were calculated.

• Transactions of the Korean Society of Mechanical Engineers
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• v.18 no.9
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• pp.2339-2348
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• 1994

The objective of this paper is to develop a solution method for the differential-algebraic equation(DAE) derived from constrained muti-body dynamic systems. Mechanical systems are often modeled as bodies and joints. Differential equations of motion are formulated for bodies. Since the bodies are connected by joint, the differential variables must satisfy the kinematic constraint equations that come from the joints. Difficulties are arised due to drift of the differential variables off the constraint equations. An optimization method is adopted to correct the drift of the differential variables. To demonstrate the efficiency of the proposed method a slider-crank mechanism is analyzed dynamically. Identical results are obtained as these from the commercial program DADS. Dynamic analysis of a High Mobility Multi-purpose Wheeled. Vehicle(HMMWV) is carried out to show the practicalism of the proposed method.

• Journal of KSNVE
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• v.2 no.3
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• pp.203-211
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• 1992

The main purpose of this paper is to present both the natural frequencies and the first critical loads of tapered columns. The ordinary differential equation governing the free vibration for tapered columns under compressive axial force is derived. Three kinds of cross sectional shape are considered in the governing equation. The Runge-Kutta method and determinant search method are used to perform the integration of the differential equation and to determine the natural frequencies, respectively. Additionally, the bisection method is used to determine the critical loads. In numerical examples, the effects of compressive axial force on the natural frequencies of tapered columns are investigated varying the end conditions. The first critical loads of tapered columns are determined on the basis of dynamic concepts. The first critical loads of tapered columns are determined on the basis of dynamic concept. The effects of cross sectional shapes are shown and some typical mode shapes are also presented.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.383-395
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• 2010

In the present paper, we construct the traveling wave solutions involving parameters of nonlinear evolution equations in the mathematical physics via the (3+1)- dimensional potential- YTSF equation, the (3+1)- dimensional generalized shallow water equation, the (3+1)- dimensional Kadomtsev- Petviashvili equation, the (3+1)- dimensional modified KdV-Zakharov- Kuznetsev equation and the (3+1)- dimensional Jimbo-Miwa equation by using a simple method which is called the ($\frac{G'}{G}$)- expansion method, where $G\;=\;G(\xi)$ satisfies a second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the travelling waves. The travelling wave solutions are expressed by hyperbolic, trigonometric and rational functions.

• Smart Structures and Systems
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• v.26 no.2
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• pp.213-226
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• 2020

In this manuscript, static and dynamic behaviors of geometrically imperfect carbon nanotubes (CNTs) subject to different types of end conditions are investigated. The Doublet Mechanics (DM) theory, which is length scale dependent theory, is used in the analysis. The Euler-Bernoulli kinematic and nonlinear mid-plane stretching effect are considered through analysis. The governing equation of imperfect CNTs is a sixth order nonlinear integro-partial-differential equation. The buckling problem is discretized via the differential-integral-quadrature method (DIQM) and then it is solved using Newton's method. The equation of linear vibration problem is discretized using DIQM and then solved as a linear eigenvalue problem to get natural frequencies and corresponding mode shapes. The DIQM results are compared with analytical ones available in the literature and excellent agreement is obtained. The numerical results are depicted to illustrate the influence of length scale parameter, imperfection amplitude and shear foundation constant on critical buckling load, post-buckling configuration and linear vibration behavior. The current model is effective in designing of NEMS, nano-sensor and nano-actuator manufactured by CNTs.