On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations
Omowo Babajide Johnson,
Longe Idowu Oluwaseun,
Osakwe Charles Nnamdi
Issue:
Volume 8, Issue 1, March 2023
Pages:
1-6
Received:
8 October 2022
Accepted:
28 November 2022
Published:
14 February 2023
Abstract: This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.
Abstract: This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann metho...
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The Extended Adjacency Indices for Several Types of Graph Operations
Feng Fu,
Bo Deng,
Hongyu Zhang
Issue:
Volume 8, Issue 1, March 2023
Pages:
7-11
Received:
4 January 2023
Accepted:
29 January 2023
Published:
14 February 2023
Abstract: Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index EA(G) is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.
Abstract: Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a ...
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Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations
Azure Isaac,
Twum Boakye Stephen,
Anas Musah,
Aloliga Golbert
Issue:
Volume 8, Issue 1, March 2023
Pages:
12-16
Received:
25 February 2023
Accepted:
3 April 2023
Published:
18 April 2023
Abstract: Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.
Abstract: Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the...
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