• Title, Summary, Keyword: initial value problems

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SERIES SOLUTIONS TO INITIAL-NEUMANN BOUNDARY VALUE PROBLEMS FOR PARABOLIC AND HYPERBOLIC EQUATIONS

  • Bougoffa, Lazhar;Al-Mazmumy, M.
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.87-97
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    • 2013
  • The purpose of this paper is to employ a new useful technique to solve the initial-Neumann boundary value problems for parabolic, hyperbolic and parabolic-hyperbolic equations and obtain a solution in form of infinite series. The results obtained indicate that this approach is indeed practical and efficient.

A NEW FIFTH-ORDER WEIGHTED RUNGE-KUTTA ALGORITHM BASED ON HERONIAN MEAN FOR INITIAL VALUE PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS

  • CHANDRU, M.;PONALAGUSAMY, R.;ALPHONSE, P.J.A.
    • Journal of applied mathematics & informatics
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    • v.35 no.1_2
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    • pp.191-204
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    • 2017
  • A new fifth-order weighted Runge-Kutta algorithm based on heronian mean for solving initial value problem in ordinary differential equations is considered in this paper. Comparisons in terms of numerical accuracy and size of the stability region between new proposed Runge-Kutta(5,5) algorithm, Runge-Kutta (5,5) based on Harmonic Mean, Runge-Kutta(5,5) based on Contra Harmonic Mean and Runge-Kutta(5,5) based on Geometric Mean are carried out as well. The problems, methods and comparison criteria are specified very carefully. Numerical experiments show that the new algorithm performs better than other three methods in solving variety of initial value problems. The error analysis is discussed and stability polynomials and regions have also been presented.

AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH A SMALL NEGATIVE SHIFT

  • Rao, R. Nageshwar;Chakravarthy, P. Pramod
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.131-145
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    • 2013
  • In this paper, we present an initial value technique for solving singularly perturbed differential difference equations with a boundary layer at one end point. Taylor's series is used to tackle the terms containing shift provided the shift is of small order of singular perturbation parameter and obtained a singularly perturbed boundary value problem. This singularly perturbed boundary value problem is replaced by a pair of initial value problems. Classical fourth order Runge-Kutta method is used to solve these initial value problems. The effect of small shift on the boundary layer solution in both the cases, i.e., the boundary layer on the left side as well as the right side is discussed by considering numerical experiments. Several numerical examples are solved to demonstate the applicability of the method.

SOLVING SECOND ORDER SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH LAYER BEHAVIOR VIA INITIAL VALUE METHOD

  • GEBEYAW, WONDWOSEN;ANDARGIE, AWOKE;ADAMU, GETACHEW
    • Journal of applied mathematics & informatics
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    • v.36 no.3_4
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    • pp.331-348
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    • 2018
  • In this paper, an initial value method for solving a class of singularly perturbed delay differential equations with layer behavior is proposed. In this approach, first the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay using Taylor series expansion. Then from the modified problem, two explicit Initial Value Problems which are independent of the perturbation parameter, ${\varepsilon}$, are produced: the reduced problem and boundary layer correction problem. Finally, these problems are solved analytically and combined to give an approximate asymptotic solution to the original problem. To demonstrate the efficiency and applicability of the proposed method three linear and one nonlinear test problems are considered. The effect of the delay on the layer behavior of the solution is also examined. It is observed that for very small ${\varepsilon}$ the present method approximates the exact solution very well.

Numerical solving of initial-value problems by Rbf basis functions

  • Gotovac, Blaz;Kozulic, Vedrana
    • Structural Engineering and Mechanics
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    • v.14 no.3
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    • pp.263-285
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    • 2002
  • This paper presents a numerical procedure for solving initial-value problems using the special functions which belong to a class of Rvachev's basis functions $R_{bf}$ based on algebraic and trigonometric polynomials. Because of infinite derivability of these functions, derivatives of all orders, required by differential equation of the problem and initial conditions, are used directly in the numerical procedure. The accuracy and stability of the proposed numerical procedure are proved on an example of a single degree of freedom system. Critical time step was also determined. An algorithm for solving multiple degree of freedom systems by the collocation method was developed. Numerical results obtained by $R_{bf}$ functions are compared with exact solutions and results obtained by the most commonly used numerical procedures for solving initial-value problems.

Numerical method to determine the elastic curve of simply supported beams of variable cross-section

  • Biro, Istvan;Cveticanin, Livija;Szuchy, Peter
    • Structural Engineering and Mechanics
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    • v.68 no.6
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    • pp.713-720
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    • 2018
  • In this paper a new numerical method to determine the elastic curve of the simply supported beams of variable cross-section is demonstrated. In general case it needs to solve linear or small nonlinear second order differential equations with prescribed boundary conditions. For numerical solution the initial values of the slope and the deflection of the end cross-section of the beam is necessary. For obtaining the initial values a lively procedure is developed: it is a special application of the shooting method because boundary value problems can be transformed into initial value problems. As a result of these transformations the initial values of the differential equations are obtained with high accuracy. Procedure is applied for calculating of elastic curve of a simply supported beam of variable cross-section. Results of these numerical procedures, analytical solution of the linearized version and finite element method are compared. It is proved that the suggested procedure yields technically accurate results.

A Chebyshev Collocation Method for Stiff Initial Value Problems and Its Stability

  • Kim, Sang-Dong;Kwon, Jong-Kyum;Piao, Xiangfan;Kim, Phil-Su
    • Kyungpook Mathematical Journal
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    • v.51 no.4
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    • pp.435-456
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    • 2011
  • The Chebyshev collocation method in [21] to solve stiff initial-value problems is generalized by using arbitrary degrees of interpolation polynomials and arbitrary collocation points. The convergence of this generalized Chebyshev collocation method is shown to be independent of the chosen collocation points. It is observed how the stability region does depend on collocation points. In particular, A-stability is shown by taking the mid points of nodes as collocation points.

A New Initial Value for Solving Redundancy Optimization Problems (중복설계 최적화문제의 새로운 초기 값 설정에 관한 연구)

  • 이도경;이근희
    • Journal of the Society of Korea Industrial and Systems Engineering
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    • v.15 no.25
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    • pp.11-14
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    • 1992
  • This paper presents a method for establishing an initial value of redundancy optimization problem to maximize system reliability of multiconstraint mixed parallel-series system. The constraints not be linear. This paper proposes a new initial value which is near to optimal solution by considering the relative median rate of the unreliability and amount of consumed resources for each subsystem. To show the efficiency of this model. numerical example and comparison with Narasimhalu is illustrated in chapter 4.

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Exponentially Fitted Error Correction Methods for Solving Initial Value Problems

  • Kim, Sang-Dong;Kim, Phil-Su
    • Kyungpook Mathematical Journal
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    • v.52 no.2
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    • pp.167-177
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    • 2012
  • In this article, we propose exponentially fitted error correction methods(EECM) which originate from the error correction methods recently developed by the authors (see [10, 11] for examples) for solving nonlinear stiff initial value problems. We reduce the computational cost of the error correction method by making a local approximation of exponential type. This exponential local approximation yields an EECM that is exponentially fitted, A-stable and L-stable, independent of the approximation scheme for the error correction. In particular, the classical explicit Runge-Kutta method for the error correction not only saves the computational cost that the error correction method requires but also gives the same convergence order as the error correction method does. Numerical evidence is provided to support the theoretical results.

Analytical solutions of in-plane static problems for non-uniform curved beams including axial and shear deformations

  • Tufekci, Ekrem;Arpaci, Alaeddin
    • Structural Engineering and Mechanics
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    • v.22 no.2
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    • pp.131-150
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    • 2006
  • Exact analytical solutions for in-plane static problems of planar curved beams with variable curvatures and variable cross-sections are derived by using the initial value method. The governing equations include the axial extension and shear deformation effects. The fundamental matrix required by the initial value method is obtained analytically. Then, the displacements, slopes and stress resultants are found analytically along the beam axis by using the fundamental matrix. The results are given in analytical forms. In order to show the advantages of the method, some examples are solved and the results are compared with the existing results in the literature. One of the advantages of the proposed method is that the high degree of statically indeterminacy adds no extra difficulty to the solution. For some examples, the deformed shape along the beam axis is determined and plotted and also the slope and stress resultants are given in tables.