Search results for: nonlinear partial differential equation.

Authors: Emin Özyılmaz

Abstract:

In this work, position vector of a time-like dual curve according to standard frame of D31 is investigated. First, it is proven that position vector of a time-like dual curve satisfies a dual vector differential equation of fourth order. The general solution of this dual vector differential equation has not yet been found. Due to this, in terms of special solutions, position vectors of some special time-like dual curves with respect to standard frame of D31 are presented.

Keywords: Classical Differential Geometry, Dual Numbers, DualFrenet Equations, Time-like Dual Curve, Position Vector, DualLorentzian Space.

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Authors: Imad Chaddad, Andrei A. Kolyshkin

Abstract:

Linear and weakly nonlinear analysis of shallow wake flows is presented in the present paper. The evolution of the most unstable linear mode is described by the complex Ginzburg-Landau equation (CGLE). The coefficients of the CGLE are calculated numerically from the solution of the corresponding linear stability problem for a one-parametric family of shallow wake flows. It is shown that the coefficients of the CGLE are not so sensitive to the variation of the base flow profile.

Keywords: Ginzburg-Landau equation, shallow wake flow, weakly nonlinear theory.

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Authors: Peter Benneth, Tsaroh N. Theophilus, Prince Benjamin

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This research work aims at studying the application of Legendre Transformation Method (LTM) to Hamilton Jacobi Bellman (HJB) equation which is an example of optimal control problem. We discuss the steps involved in modelling the HJB equation as it relates to mathematical finance by applying the Ito’s lemma and maximum principle theorem. By applying the LTM and dual theory, the resultant HJB equation is transformed to a linear Partial Differential Equation (PDE). Also, the Optimal Investment Strategy (OIS) and the optimal value function were obtained under the exponential utility function. Furthermore, some numerical results were also presented with observations that the OIS under exponential utility is directly proportional to the appreciation rate of the risky asset and inversely proportional to the instantaneous volatility, predetermined interest rate, risk averse coefficient. Finally, it was observed that the optimal fund size is an increasing function of the risk free interest rate. This result is consistent with some existing results.

Keywords: Legendre transformation method, Optimal investment strategy, Ito’s lemma, Hamilton Jacobi Bellman equation, Geometric Brownian motion, financial market.

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Authors: Taher Lotfi , Parisa Bakhtiari , Katayoun Mahdiani , Mehdi Salimi

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In this paper, verified extension of the Ostrowski method which calculates the enclosure solutions of a given nonlinear equation is introduced. Also, error analysis and convergence will be discussed. Some implemented examples with INTLAB are also included to illustrate the validity and applicability of the scheme.

Keywords: Iinterval analysis, nonlinear equations, Ostrowski method.

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Authors: M.A.Behrang, M. Ghalambaz, E. Assareh, A.R. Noghrehabadi

Abstract:

Fluid flow and heat transfer of vertical full cone embedded in porous media is studied in this paper. Nonlinear differential equation arising from similarity solution of inverted cone (subjected to wall temperature boundary conditions) embedded in porous medium is solved using a hybrid neural network- particle swarm optimization method. To aim this purpose, a trial solution of the differential equation is defined as sum of two parts. The first part satisfies the initial/ boundary conditions and does contain an adjustable parameter and the second part which is constructed so as not to affect the initial/boundary conditions and involves adjustable parameters (the weights and biases) for a multi-layer perceptron neural network. Particle swarm optimization (PSO) is applied to find adjustable parameters of trial solution (in first and second part). The obtained solution in comparison with the numerical ones represents a remarkable accuracy.

Keywords: Porous Media, Ordinary Differential Equations (ODE), Particle Swarm Optimization (PSO), Neural Network (NN).

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Authors: Xiguang Li

Abstract:

In this paper , by using fixed point theorem , upper and lower solution-s method and monotone iterative technique , we prove the existence of maximum and minimum solutions of differential equations with delay , which improved and generalize the result of related paper.

Keywords: Banach space, boundary value problem, differential equation, delay.

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Authors: P.-W. Tsai, W.-L. Hong, C.-W. Chen, C.-Y. Chen

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In this paper, we present a neural-network (NN) based approach to represent a nonlinear Tagagi-Sugeno (T-S) system. A linear differential inclusion (LDI) state-space representation is utilized to deal with the NN models. Taking advantage of the LDI representation, the stability conditions and controller design are derived for a class of nonlinear structural systems. Moreover, the concept of utilizing the Parallel Particle Swarm Optimization (PPSO) algorithm to solve the common P matrix under the stability criteria is given in this paper.

Keywords: Lyapunov Stability, Parallel Particle Swarm Optimization, Linear Differential Inclusion, Artificial Intelligence.

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Authors: Mohamed M. Mousa, Aidarkhan Kaltayev

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Based on the homotopy perturbation method (HPM) and Padé approximants (PA), approximate and exact solutions are obtained for cubic Boussinesq and modified Boussinesq equations. The obtained solutions contain solitary waves, rational solutions. HPM is used for analytic treatment to those equations and PA for increasing the convergence region of the HPM analytical solution. The results reveal that the HPM with the enhancement of PA is a very effective, convenient and quite accurate to such types of partial differential equations.

Keywords: Homotopy perturbation method, Padé approximants, cubic Boussinesq equation, modified Boussinesq equation.

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Authors: A. Thabet, G. B. H. Frej, M. Boutayeb

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This paper presents a design of dynamic feedback control based on observer for a class of large scale Lipschitz nonlinear systems. The use of Differential Mean Value Theorem (DMVT) is to introduce a general condition on the nonlinear functions. To ensure asymptotic stability, sufficient conditions are expressed in terms of linear matrix inequalities (LMIs). High performances are shown through real time implementation with ARDUINO Duemilanove board to the one-link flexible joint robot.

Keywords: Feedback stabilization, DMVT, Lipschitz nonlinear systems, nonlinear observer, real time implementation.

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Authors: Yanling Zhu, Kai Wang

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By means of Mawhin’s continuation theorem, we study a kind of third-order p-Laplacian functional differential equation with distributed delay in the form: ϕp(x (t)) = g  t,  0 −τ x(t + s) dα(s)  + e(t), some criteria to guarantee the existence of periodic solutions are obtained.

Keywords: p–Laplacian, distributed delay, periodic solution, Mawhin's continuation theorem.

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Authors: Aomar Anane, Omar Chakrone, Loubna Moutaouekkil

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As p-Laplacian equations have been widely applied in field of the fluid mechanics and nonlinear elastic mechanics, it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p-Laplacian. By using Mawhin’s continuation theorem, we study the existence of periodic solutions for p-Laplacian neutral Rayleigh equation (ϕp(x(t)−c(t)x(t − r))) + f(x(t)) + g1(x(t − τ1(t, |x|∞))) + β(t)g2(x(t − τ2(t, |x|∞))) = e(t), It is meaningful that the functions c(t) and β(t) are allowed to change signs in this paper, which are different from the corresponding ones of known literature.

Keywords: periodic solution, neutral Rayleigh equation, variable sign, Deviating argument, p-Laplacian, Mawhin’s continuation.

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Authors: Bhanu Gupta, Sanjay K. Srivastava

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In this paper, we shall present sufficient conditions for the ψ-exponential stability of a class of nonlinear impulsive differential equations. We use the Lyapunov method with functions that are not necessarily differentiable. In the last section, we give some examples to support our theoretical results.

Keywords: Exponential stability, globally exponential stability, impulsive differential equations, Lyapunov function, ψ-stability.

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Authors: Ahmad Fahim Nasiry, Shigeo Honma

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We examine two-dimensional oil displacement by water in a petroleum reservoir. The pore fluid is immiscible, and the porous media is homogenous and isotropic in the horizontal direction. Buckley-Leverett theory and a combination of Laplacian and Darcy’s law are used to study the fluid flow through porous media, and the Laplacian that defines the dispersion and diffusion of fluid in the sand using heavy oil is discussed. The reservoir is homogenous in the horizontal direction, as expressed by the partial differential equation. Two main factors which are observed are the water saturation and pressure distribution in the reservoir, and they are evaluated for predicting oil recovery in two dimensions by a physical and mathematical simulation model. We review the numerical simulation that solves difficult partial differential reservoir equations. Based on the numerical simulations, the saturation and pressure equations are calculated by the iterative alternating direction implicit method and the iterative alternating direction explicit method, respectively, according to the finite difference assumption. However, to understand the displacement of oil by water and the amount of water dispersion in the reservoir better, an interpolated contour line of the water distribution of the five-spot pattern, that provides an approximate solution which agrees well with the experimental results, is also presented. Finally, a computer program is developed to calculate the equation for pressure and water saturation and to draw the pressure contour line and water distribution contour line for the reservoir.

Keywords: Numerical simulation, immiscible, finite difference, IADI, IADE, waterflooding.

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Authors: Sarun Phibanchon

Abstract:

The solitary wave solution of the quadratic nonlinear Schrdinger equation is determined by the iterative method called Petviashvili method. This solution is also used for the initial condition for the time evolution to study the stability analysis. The spectral method is applied for the time evolution.

Keywords: soliton, iterative method, spectral method, plasma

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Authors: Nisha Goyal, R.K. Gupta

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This paper presents a new function expansion method for finding traveling wave solutions of a nonlinear equations and calls it the G G -expansion method, given by Wang et al recently. As an application of this new method, we study the well-known Sawada-Kotera-Kadomtsev-Petviashivili equation and Bogoyavlensky-Konoplechenko equation. With two new expansions, general types of soliton solutions and periodic solutions for these two equations are obtained.

Keywords: Sawada-Kotera-Kadomtsev-Petviashivili equation, Bogoyavlensky-Konoplechenko equation,

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Authors: Roslinda Nazar, Amin Noor, Khamisah Jafar, Ioan Pop

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In this paper, the steady laminar three-dimensional boundary layer flow and heat transfer of a copper (Cu)-water nanofluid in the vicinity of a permeable shrinking flat surface in an otherwise quiescent fluid is studied. The nanofluid mathematical model in which the effect of the nanoparticle volume fraction is taken into account is considered. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations using a similarity transformation which is then solved numerically using the function bvp4c from Matlab. Dual solutions (upper and lower branch solutions) are found for the similarity boundary layer equations for a certain range of the suction parameter. A stability analysis has been performed to show which branch solutions are stable and physically realizable. The numerical results for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles are obtained, presented and discussed in detail for a range of various governing parameters.

Keywords: Heat Transfer, Nanofluid, Shrinking Surface, Stability Analysis, Three-Dimensional Flow.

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Authors: Banaja Mohanty, Prakash Kumar Hota

Abstract:

This paper presents a differential evolution algorithm to design a robust PI and PID controllers for Load Frequency Control (LFC) of nonlinear interconnected power systems considering the boiler dynamics, Governor Dead Band (GDB), Generation Rate Constraint (GRC). Differential evolution algorithm is employed to search for the optimal controller parameters. The proposed method easily copes of with nonlinear constraints. Further the proposed controller is simple, effective and can ensure the desirable overall system performance. The superiority of the proposed approach has been shown by comparing the results with published fuzzy logic controller for the same power systems. The comparison is done using various performance measures like overshoot, settling time and standard error criteria of frequency and tie-line power deviation following a 1% step load perturbation in hydro area. It is noticed that, the dynamic performance of proposed controller is better than fuzzy logic controller. Furthermore, it is also seen that the proposed system is robust and is not affected by change in the system parameters.

Keywords: Automatic Generation control (AGC), Generation Rate Constraint (GRC), Governor Dead Band (GDB), Differential Evolution (DE)

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Authors: Azizollah Khormali, Seyyed Shahab Tabatabaee Moradi, Dmitry Petrakov

Abstract:

One of the most important parameters in petroleum reservoirs is the pressure distribution along the reservoir, as the pressure varies with the time and location. A popular method to determine the pressure distribution in a reservoir in the unsteady state regime of flow is applying Darcy’s equation and solving this equation numerically. The numerical simulation of reservoirs is based on these numerical solutions of different partial differential equations (PDEs) representing the multiphase flow of fluids. Pressure profile has obtained in a one dimensional system solving Darcy’s equation explicitly. Changes of pressure profile in three situations are investigated in this work. These situations include section length changes, step time changes and time approach to infinity. The effects of these changes in pressure profile are shown and discussed in the paper.

Keywords: Explicit solution, Numerical simulation, Petroleum reservoir, Pressure distribution.

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Authors: Javad Abdalkhani

Abstract:

Convergence of power series solutions for a class of non-linear Abel type equations, including an equation that arises in nonlinear cooling of semi-infinite rods, is very slow inside their small radius of convergence. Beyond that the corresponding power series are wildly divergent. Implementation of nonlinear sequence transformation allow effortless evaluation of these power series on very large intervals..

Keywords: Nonlinear transformation, Abel Volterra Equations, Mathematica

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Authors: Anjali Verma, Ram Jiwari, Jitender Kumar

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This paper presents a new function expansion method for finding traveling wave solution of a non-linear equation and calls it the (G'/G)-expansion method. The shallow water wave equation is reduced to a non linear ordinary differential equation by using a simple transformation. As a result the traveling wave solutions of shallow water wave equation are expressed in three forms: hyperbolic solutions, trigonometric solutions and rational solutions.

Keywords: Shallow water wave equation, Exact solutions, (G'/G) expansion method.

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Authors: Marziyeh Halimi, Roushanak Lotfikar, Simin Mansouri Borojeni

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In this paper we considered the Neumann problem for the fourth order differential equation. First we define the weighted Sobolev space 2 Wα and generalized solution for this equation. Then we consider the existence and uniqueness of the generalized solution, as well as give the description of the spectrum and of the domain of definition of the corresponding operator.

Keywords: Neumann problem, weighted Sobolev spaces, generalized solution, spectrum of linear operators.2000 mathematic subject classification: 34A05, 34A30.

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Authors: Irina Eglite, Andrei A. Kolyshkin

Abstract:

Method of multiple scales is used in the paper in order to derive an amplitude evolution equation for the most unstable mode from two-dimensional shallow water equations under the rigid-lid assumption. It is assumed that shallow mixing layer is slightly curved in the longitudinal direction and contains small particles. Dynamic interaction between carrier fluid and particles is neglected. It is shown that the evolution equation is the complex Ginzburg-Landau equation. Explicit formulas for the computation of the coefficients of the equation are obtained.

Keywords: Shallow water equations, mixing layer, weakly nonlinear analysis, Ginzburg-Landau equation

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Authors: Janak Raj Sharma, Rajni Sharma

Abstract:

Based on Traub-s methods for solving nonlinear equation f(x) = 0, we develop two families of third-order methods for solving system of nonlinear equations F(x) = 0. The families include well-known existing methods as special cases. The stability is corroborated by numerical results. Comparison with well-known methods shows that the present methods are robust. These higher order methods may be very useful in the numerical applications requiring high precision in their computations because these methods yield a clear reduction in number of iterations.

Keywords: Nonlinear equations and systems, Newton's method, fixed point iteration, order of convergence.

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Authors: Hesham A. Elkaranshawy, Amr M. Abdelrazek, Hosam M. Ezzat

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The system of ordinary nonlinear differential equations describing sliding velocity during impact with friction for a three-dimensional rigid-multibody system is developed. No analytical solutions have been obtained before for this highly nonlinear system. Hence, a power series solution is proposed. Since the validity of this solution is limited to its convergence zone, a suitable time step is chosen and at the end of it a new series solution is constructed. For a case study, the trajectory of the sliding velocity using the proposed method is built using 6 time steps, which coincides with a Runge- Kutta solution using 38 time steps.

Keywords: Impact with friction, nonlinear ordinary differential equations, power series solutions, rough collision.

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Authors: T. Yamaguchi, M. Watanabe, M. Sasajima, C. Yuan, S. Maruyama, T. B. Ibrahim, H. Tomita

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This paper deals with nonlinear vibration analysis using finite element method for frame structures consisting of elastic and viscoelastic damping layers supported by multiple nonlinear concentrated springs with hysteresis damping. The frame is supported by four nonlinear concentrated springs near the four corners. The restoring forces of the springs have cubic non-linearity and linear component of the nonlinear springs has complex quantity to represent linear hysteresis damping. The damping layer of the frame structures has complex modulus of elasticity. Further, the discretized equations in physical coordinate are transformed into the nonlinear ordinary coupled differential equations using normal coordinate corresponding to linear natural modes. Comparing shares of strain energy of the elastic frame, the damping layer and the springs, we evaluate the influences of the damping couplings on the linear and nonlinear impact responses. We also investigate influences of damping changed by stiffness of the elastic frame on the nonlinear coupling in the damped impact responses.

Keywords: Dynamic response, Nonlinear impact response, Finite Element analysis, Numerical analysis.

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Authors: Vikas Tewari, K.R. Pardasani

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Calcium is very important for communication among the neurons. It is vital in a number of cell processes such as secretion, cell movement, cell differentiation. To reduce the system of reactiondiffusion equations of [Ca2+] into a single equation, two theories have been proposed one is excess buffer approximation (EBA) other is rapid buffer approximation (RBA). The RBA is more realistic than the EBA as it considers both the mobile and stationary endogenous buffers. It is valid near the mouth of the channel. In this work we have studied the effects of different types of buffers on calcium diffusion under RBA. The novel thing studied is the effect of sodium ions on calcium diffusion. The model has been made realistic by considering factors such as variable [Ca2+], [Na+] sources, sodium-calcium exchange protein(NCX), Sarcolemmal Calcium ATPase pump. The proposed mathematical leads to a system of partial differential equations which has been solved numerically to study the relationships between different parameters such as buffer concentration, buffer disassociation rate, calcium permeability. We have used Forward Time Centred Space (FTCS) approach to solve the system of partial differential equations.

Keywords: rapid buffer approximation, sodium-calcium exchangeprotein, Sarcolemmal Calcium ATPase pump, buffer disassociationrate, forward time centred space.

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Authors: Hassan Manouzi

Abstract:

We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.

Keywords: Least squares, Wick product, SPDEs, finite element, Wiener chaos expansion, gradient method.

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Authors: Sanjeeb Kumar Kar

Abstract:

The optimal control problem of a linear distributed parameter system is studied via shifted Legendre polynomials (SLPs) in this paper. The partial differential equation, representing the linear distributed parameter system, is decomposed into an n - set of ordinary differential equations, the optimal control problem is transformed into a two-point boundary value problem, and the twopoint boundary value problem is reduced to an initial value problem by using SLPs. A recursive algorithm for evaluating optimal control input and output trajectory is developed. The proposed algorithm is computationally simple. An illustrative example is given to show the simplicity of the proposed approach.

Keywords: Optimal control, linear systems, distributed parametersystems, Legendre polynomials.

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Authors: Faheem Ahmed, Fareed Ahmed, Yongheng Guo, Yong Yang

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This paper deals with a high-order accurate Runge Kutta Discontinuous Galerkin (RKDG) method for the numerical solution of the wave equation, which is one of the simple case of a linear hyperbolic partial differential equation. Nodal DG method is used for a finite element space discretization in 'x' by discontinuous approximations. This method combines mainly two key ideas which are based on the finite volume and finite element methods. The physics of wave propagation being accounted for by means of Riemann problems and accuracy is obtained by means of high-order polynomial approximations within the elements. High order accurate Low Storage Explicit Runge Kutta (LSERK) method is used for temporal discretization in 't' that allows the method to be nonlinearly stable regardless of its accuracy. The resulting RKDG methods are stable and high-order accurate. The L1 ,L2 and L∞ error norm analysis shows that the scheme is highly accurate and effective. Hence, the method is well suited to achieve high order accurate solution for the scalar wave equation and other hyperbolic equations.

Keywords: Nodal Discontinuous Galerkin Method, RKDG, Scalar Wave Equation, LSERK

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Authors: Jaipong Kasemsuwan

Abstract:

This paper presents the finite difference scheme and the numerical simulation of suspended string. The vibration solutions when the various external forces are taken into account are obtained and compared with the solutions without external force. In addition, we also investigate how the external forces and their powers and coefficients affect the amplitude of vibration.

Keywords: Nonlinear external forces, Numerical simulation, Suspended string equation.

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