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On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)
Mohammad Roknujjaman,
Mohammad Asaduzzaman
Issue:
Volume 6, Issue 1, February 2018
Pages:
1-7
Received:
28 December 2017
Accepted:
16 January 2018
Published:
12 February 2018
Abstract: In this paper we used the method of lines (MOL) as a solution procedure for solving partial differential equation (PDE). The range of applications of the MOL has increased dramatically in the last few years; nevertheless, there is no introductory to initiate a beginner to the method. This Paper illustrates the application of the MOL using Crank-Nicholson method (CNM) for numerical solution of PDE together with initial condition and Dirichlet’s Boundary Condition. The implementation of this solutions is done using Microsoft office excel worksheet or spreadsheet, Matlab programming language. Finally, here we analysis the particular solution and numerical solution of Laplace equation obtained by MOL along with CNM.
Abstract: In this paper we used the method of lines (MOL) as a solution procedure for solving partial differential equation (PDE). The range of applications of the MOL has increased dramatically in the last few years; nevertheless, there is no introductory to initiate a beginner to the method. This Paper illustrates the application of the MOL using Crank-Nic...
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Mathematical Model of Corrective Maintenance Based on Operability Checks for Safety Critical Systems
Issue:
Volume 6, Issue 1, February 2018
Pages:
8-14
Received:
20 January 2018
Accepted:
1 February 2018
Published:
5 March 2018
Abstract: Maintenance based on equipment operability checks is widely used for technical systems of various physical nature. For commercial and military aircraft such checks are carried-out after a certain amount of time according to specific maintenance programs. Therefore, great attention in the research literature is paid to the mathematical modeling of maintenance on the basis of equipment operability checks. In this study, a mathematical model of corrective maintenance with operability checks at discrete times for the safety critical systems is considered. The criterion of the corrective maintenance effectiveness is proposed to provide a given level of operational reliability with minimum maintenance costs. A finite time interval is considered for modeling the moments of the system operability checks. The graph of decision making is analyzed for imperfect operability checks and the probabilities of possible decisions are determined. Analytical equations for the operational reliability and expected maintenance costs are derived for an arbitrary distribution of time to failure. The criteria of determining optimal policies of sequential checks are formulated. Numerical examples illustrate the developed theory. For the first time it has been shown that conditional probabilities of correct and incorrect decisions when checking system operability are dependent on the time of failure and parameters of the degradation model. Numerical calculations have shown that in the case of mixing deteriorating systems with different initial time points of operation, the interval between operability checks converges to a constant periodicity.
Abstract: Maintenance based on equipment operability checks is widely used for technical systems of various physical nature. For commercial and military aircraft such checks are carried-out after a certain amount of time according to specific maintenance programs. Therefore, great attention in the research literature is paid to the mathematical modeling of m...
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Some Paradoxes of Mathematical Theory of Continues Mechanics
Issue:
Volume 6, Issue 1, February 2018
Pages:
15-19
Received:
11 January 2018
Accepted:
29 January 2018
Published:
7 March 2018
Abstract: The classic theory of continuum mechanics does not preserve the continuity of the environment due to the use of the conditions of equilibrium of forces and the symmetry of the stress tensor. We used many unreasonable mathematical approximations when by the Boltzmann equation is solved to describe the equations of continuum mechanics. The paper presents an analysis of mathematical approximations underlying description in different environments, and new models, to avoid the resulting misunderstandings. For rarefied gas the self-diffusion and thermo-diffusion which were foretold by S. V. Vallander are obtained from kinetic theory.
Abstract: The classic theory of continuum mechanics does not preserve the continuity of the environment due to the use of the conditions of equilibrium of forces and the symmetry of the stress tensor. We used many unreasonable mathematical approximations when by the Boltzmann equation is solved to describe the equations of continuum mechanics. The paper pres...
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Hamiltonicity of Mycielski Graphs
Shuting Cheng,
Dan Wang,
Xiaoping Liu
Issue:
Volume 6, Issue 1, February 2018
Pages:
20-22
Received:
28 January 2018
Accepted:
16 February 2018
Published:
16 March 2018
Abstract: Fisher, McKenna, and Boyer showed that if a graph G is hamiltonian, then its Mycielski graph μ(G) is hamitonian. In this note, it was shown that for a bipartite graph G, if its mycielski graph μ(G) is hamiltonian, then G has a Hamilton path.
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On the Planarity of G++−
Issue:
Volume 6, Issue 1, February 2018
Pages:
23-27
Received:
10 February 2018
Accepted:
1 March 2018
Published:
22 March 2018
Abstract: Let G be a simple graph. The transformation graph G++− of G is the graph with vertex set V (G) ∪ E (G) in which the vertex x and y are joined by an edge if and only if the following condition holds:(i) x, y ∈ V (G) and x and y are adjacent in G, (ii) x, y ∈ E (G), and x and y are adjacent in G, (iii) one of x and y is in V (G) and the other is in E (G), and they are not incident in G. In this paper, it is shown G++− is planar if and only if |E (G)| ≤ 2 or G is isomorphic to one of the following graphs: C3, C3 + K1, P4, P4 + K1, P3 + K2, P3 + K2 + K1, K1,3, K1,3 +K1, 3K2, 3K2 + K1, 3K2 + 2K1, C4, C4 + K1.
Abstract: Let G be a simple graph. The transformation graph G++− of G is the graph with vertex set V (G) ∪ E (G) in which the vertex x and y are joined by an edge if and only if the following condition holds:(i) x, y ∈ V (G) and x and y are adjacent in G, (ii) x, y ∈ E (G), and x and y are adjacent in G, (iii) one of x and y is in V (G) and the other is in E...
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