scholarly journals Free Axial Vibrations of Non-Uniform Rods

2011 ◽  
Vol 2-3 ◽  
pp. 882-889
Author(s):  
Shu Qi Guo ◽  
Shao Pu Yang
Keyword(s):  
Exact Solution ◽  
Material Properties ◽  
Taylor Expansion ◽  
Series Solution ◽  
Wkb Method ◽  
Kummer Functions ◽  
Axial Vibrations ◽  
Special Case

Free axial vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the pro-posed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the proposed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.

scholarly journals Series Solution for Single and System of Non-linear Volterra Integral Equations

2020 ◽  
Vol 4 (2) ◽  
pp. 6-8
Author(s):  
Narmeen N. Nadir
Keyword(s):  
Exact Solution ◽  
Integral Equations ◽  
Taylor Expansion ◽  
Series Solution ◽  
Volterra Integral Equations ◽  
Non Linear ◽  
Linear Volterra Integral Equations

In this paper, Taylor expansion has been used for solving non-linear Volterra integral equations (VIEs) of the second kind. This method allows us to overcome the difficulty caused by integrals and non-linearity; also, it has more precise and rapidly convergent to the exact solution. Two examples are presented for illustrate the performance of this method.

On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales

1968 ◽  
Vol 34 (1) ◽  
pp. 145-158 ◽  
Author(s):  
K. Kuen Tam
Keyword(s):  
Exact Solution ◽  
Velocity Profile ◽  
Asymptotic Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Order Equation ◽  
Zeroth Order ◽  
Independent Variables ◽  
Sommerfeld Equation ◽  
Special Case

The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.

scholarly journals Satisfying positivity requirement in the Beyond Complex Langevin approach

2018 ◽  
Vol 175 ◽  
pp. 11026 ◽  
Author(s):  
Adam Wyrzykowski ◽  
Błażej Ruba Ruba
Keyword(s):  
Exact Solution ◽  
Approximate Solutions ◽  
Groebner Basis ◽  
Matching Conditions ◽  
Quadratic Equations ◽  
Positive Distribution ◽  
Langevin Approach ◽  
Basis Method ◽  
Special Case ◽  
Complex Density

The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.

scholarly journals Matrix-valued Bratu equation and the exact solution of its initial value problem

2020 ◽  
pp. 1950007
Author(s):  
Hiroto Inoue
Keyword(s):  
Exact Solution ◽  
Power Series ◽  
Initial Value Problem ◽  
Series Solution ◽  
Power Series Solution ◽  
Matrix Solution ◽  
Initial Value ◽  
Bratu Equation ◽  
Exponential Matrix

A matrix-valued extension of the Bratu equation is defined. For its initial value problem, the exponential matrix solution and power series solution are provided.

scholarly journals An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 45
Author(s):  
Wensheng Jia ◽  
Xiaoling Qiu ◽  
Dingtao Peng
Keyword(s):  
Exact Solution ◽  
Bounded Rationality ◽  
Equilibrium Problems ◽  
Approximation Theorem ◽  
Optimization Problems ◽  
Vector Equilibrium Problems ◽  
Special Cases ◽  
Full Rationality ◽  
Vector Equilibrium ◽  
Special Case

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.

scholarly journals Computational Methods for Solving Linear Fuzzy Volterra Integral Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jihan Hamaydi ◽  
Naji Qatanani
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Taylor Expansion ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Iteration Methods ◽  
Good Agreement

Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.

A Theory of Lubrication With Turbulent Flow and Its Application to Slider Bearings

1960 ◽  
Vol 27 (1) ◽  
pp. 1-4 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Exact Solution ◽  
Reynolds Number ◽  
Turbulent Flow ◽  
Closed Form ◽  
Governing Equation ◽  
Reynolds Equation ◽  
Three Dimensions ◽  
Slider Bearing ◽  
Numerical Example ◽  
Special Case

The governing equation of turbulent lubrication in three dimensions, equivalent to the Reynolds equation of laminar lubrication, is derived. The problem of a slider bearing with no side leakage is then analyzed. An exact solution is found in closed form. Bearing characteristics are also established. It is found that the Reynolds number is an important parameter in the problem of turbulent lubrication. Furthermore, it is shown that the laminar lubrication may be considered as the special case of the present study. A numerical example is also included.

scholarly journals Solving Fuzzy Differential Equations by Using Power Series

2020 ◽  
pp. 92-107
Author(s):  
Rasha H. Ibraheem
Keyword(s):  
Approximate Solution ◽  
Power Series ◽  
Differential Equations ◽  
Taylor Expansion ◽  
Series Solution ◽  
Convergent Series ◽  
Initial Value ◽  
Third Order ◽  
Technical Accuracy ◽  
Made In

In this paper, the series solution is applied to solve third order fuzzy differential equations with a fuzzy initial value. The proposed method applies Taylor expansion in solving the system and the approximate solution of the problem which is calculated in the form of a rapid convergent series; some definitions and theorems are reviewed as a basis in solving fuzzy differential equations. An example is applied to illustrate the proposed technical accuracy. Also, a comparison between the obtained results is made, in addition to the application of the crisp solution, when the-level equals one.

scholarly journals Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2231
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Power Law ◽  
Dynamic Processes ◽  
Nonlinear Fractional Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
Special Case

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

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