scholarly journals Analytic-Numerical Solution of Random Boundary Value Heat Problems in a Semi-Infinite Bar

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
M.-C. Casabán ◽  
J.-C. Cortés ◽  
B. García-Mora ◽  
L. Jódar
Keyword(s):  
Stochastic Process ◽  
Temperature Distribution ◽  
Numerical Solution ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mean Square ◽  
Random Boundary ◽  
Numerical Examples

This paper deals with the analytic-numerical solution of random heat problems for the temperature distribution in a semi-infinite bar with different boundary value conditions. We apply a random Fourier sine and cosine transform mean square approach. Random operational mean square calculus is developed for the introduced transforms. Using previous results about random ordinary differential equations, a closed form solution stochastic process is firstly obtained. Then, expectation and variance are computed. Illustrative numerical examples are included.

scholarly journals A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 631 ◽  
Author(s):  
Yong-Sheng Lian ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Xiao-Ting He ◽  
Zhou-Lian Zheng
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Circular Membrane ◽  
Computational Accuracy ◽  
Power Series Method ◽  
Membrane Problem ◽  
Geometric Equation

In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption of the membrane was given up. In particular, a more effective geometric equation was, for the first time, established to replace the classic one, and finally the resulting new boundary value problem due to the improvement of geometric equation was successfully solved by the power series method. The conducted numerical example indicates that the closed-form solution presented in this study has higher computational accuracy in comparison with the existing solutions of the well-known Föppl–Hencky membrane problem. In addition, some important issues were discussed, such as the difference between membrane problems and thin plate problems, reasonable approximation or assumption during establishing geometric equations, and the contribution of reducing approximations or relaxing assumptions to the improvement of the computational accuracy and applicability of a solution. Finally, some opinions on the follow-up work for the well-known Föppl–Hencky membrane were presented.

Structure design and kinematics of a robot manipulator

Robotica ◽  
1988 ◽  
Vol 6 (4) ◽  
pp. 299-309 ◽  
Author(s):  
Kesheng Wang ◽  
Terje K. Lien
Keyword(s):  
Numerical Solution ◽  
Closed Form ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Robot Manipulator ◽  
Closed Form Solution ◽  
Form Solution ◽  
Structure Design ◽  
General Algorithm ◽  
6 Degrees Of Freedom

SUMMARYIn this paper we show that a robot manipulator with 6 degrees of freedom can be separated into two parts: arm with the first three joints for major positioning and wrist with the last three joints for major orienting. We propose 5 arms and 2 wrists as basic construction for commercially robot manipulators. This kind of simplification can lead to a general algorithm of inverse kinematics for the corresponding configuration of different combinations of arm and wrist. The approaches for numerical solution and closed form solution presented in this paper are very efficient and easy for calculating the inverse kinematics of robot manipulator.

scholarly journals A Closed-Form Solution for the Boundary Value Problem of Gas Pressurized Circular Membranes in Contact with Frictionless Rigid Plates

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1017
Author(s):  
Dong Mei ◽  
Jun-Yi Sun ◽  
Zhi-Hang Zhao ◽  
Xiao-Ting He
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Gas Pressure ◽  
Circular Membrane ◽  
Rigid Plate ◽  
Pressure Loading ◽  
Numerical Example

In this paper, the static problem of equilibrium of contact between an axisymmetric deflected circular membrane and a frictionless rigid plate was analytically solved, where an initially flat circular membrane is fixed on its periphery and pressurized on one side by gas such that it comes into contact with a frictionless rigid plate, resulting in a restriction on the maximum deflection of the deflected circular membrane. The power series method was employed to solve the boundary value problem of the resulting nonlinear differential equation, and a closed-form solution of the problem addressed here was presented. The difference between the axisymmetric deformation caused by gas pressure loading and that caused by gravity loading was investigated. In order to compare the presented solution applying to gas pressure loading with the existing solution applying to gravity loading, a numerical example was conducted. The result of the conducted numerical example shows that the two solutions agree basically closely for membranes lightly loaded and diverge as the external loads intensify.

Apparently First Closed-Form Solution for Frequencies of Deterministically and/or Stochastically Inhomogeneous Simply Supported Beams

2000 ◽  
Vol 68 (2) ◽  
pp. 176-185 ◽  
Author(s):  
S. Candan ◽  
I. Elishakoff
Keyword(s):  
Closed Form ◽  
Infinite Number ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Benchmark Solutions

An infinite number of closed-form solutions is reported for a deterministically or stochastically nonhomogeneous beam, for both natural frequencies and reliabilities, for specialized cases. These solutions may prove useful as benchmark solutions. Numerical examples are evaluated.

scholarly journals Warping Effects in Strongly Perturbed Metrics

2020 ◽  
Author(s):  
Marco Frasca ◽  
Riccardo Maria Liberati ◽  
Massimiliano Rossi
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Symmetric Case ◽  
Warp Factor ◽  
Leading Order ◽  
Gradient Expansion ◽  
Numerical Examples

A technique devised some years ago permits to study a theory in a regime of strong perturbations. This translate into a gradient expansion that, at the leading order, can recover the BKL solution. We solve exactly the leading order equations in a spherical symmetric case and we show that the 4-velocity in such a case is multiplied by an exponential warp factor when the perturbation is properly applied. This factor is always greater than one. We will give a closed form solution of this factor for a simple case. Some numerical examples are also given.

scholarly journals Warping Effects in Strongly Perturbed Metrics

Physics ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 665-678
Author(s):  
Marco Frasca ◽  
Riccardo Maria Liberati ◽  
Massimiliano Rossi
Keyword(s):  
General Relativity ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Symmetric Case ◽  
Einstein Equations ◽  
Time Dependent ◽  
Warp Factor ◽  
Leading Order ◽  
Numerical Examples

A technique devised some years ago permits us to develop a theory regarding a regime of strong perturbations. This translates into a gradient expansion that, at the leading order, can recover the Belinsky-Kalathnikov-Lifshitz solution for general relativity. We solve exactly the leading order Einstein equations in a spherical symmetric case, assuming a Schwarzschild metric under the effect of a time-dependent perturbation, and we show that the 4-velocity in such a case is multiplied by an exponential warp factor when the perturbation is properly applied. This factor is always greater than one. We will give a closed form solution of this factor for a simple case. Some numerical examples are also given.

scholarly journals Closed form solution to some mixed boundary value problems for a charged sphere

1987 ◽  
Vol 28 (3) ◽  
pp. 296-309 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Closed Form ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Spherical Segment ◽  
Charged Sphere ◽  
Elementary Functions ◽  
Arbitrary Potential ◽  
Arbitrary Charge

AbstractA new method is described which allows an exact solution in a closed form to the following non-axisymmetric mixed boundary-value problem for a charged sphere: arbitrary potential values are given at the surface of a spherical segment while an arbitrary charge distribution is prescribed on the rest of the sphere. The method is founded on a new integral representation of the kernel of the governing integral equation. Several examples are considered. All the results are expressed in elementary functions. Some further applications of the method are discussed. No similar result seems to have been published previously.

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