Parabolic Methods for One Dimensional Advection-Diffusion Type Equation and Application to Burger Equation
Kedir Aliyi Koroche Kedir Aliyi Koroche
Issue:
Volume 7, Issue 2, June 2021
Pages:
40-52
Received:
26 March 2021
Accepted:
21 May 2021
Published:
27 May 2021
Abstract: In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given advection-diffusion type equation and converting it into the system of first-order ordinary differential equation in terms of temporarily variable. Next, by using Taylor series expansion we discretized the obtained system of ordinary differential equation and obtain the central finite difference equation. Then using this difference equation, the given advection-diffusion type equation is solved by using the parabolic method at each specified grid point. To validate the applicability of the proposed method, four model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm L2 and maximum absolute error norm L∞ and the local behavior of the solution is captured exactly. Numerical, exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root means square error norm L2 and maximum absolute error norm L∞ presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving advection-diffusion type equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution. Hence the proposed method is accruable to solve the advection-diffusion type equation.
Abstract: In this paper, the piecewise parabolic method is presented for solving the one-dimensional advection-diffusion type equation and its application to the burger equation. First, the given solution domain is discretized by using a uniform Discretization grid point. Next by applying the integration in terms of spatial variable, we discretized the given...
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Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension
Geleta Kinkino Meyu,
Kedir Aliyi Koriche
Issue:
Volume 7, Issue 2, June 2021
Pages:
53-61
Received:
25 January 2021
Accepted:
7 June 2021
Published:
2 July 2021
Abstract: The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between material bodies as result of temperature difference. Thus, there are three modes of heat transfer, i.e., conduction, radiation and convection. Conduction can be steady state heat conduction, or unsteady state heat conduction. If the system is in steady state, temperature doesn’t vary with time, but if the system in unsteady state temperature may varies with time. However, if the temperature of material is changing with time or if there are heat sources or sinks within the material the situation is more complex. So, rather than to escape all problem, we are targeted to solve one problem of heat equation in one dimension. The treatment was from both the analytical and the numerical view point, so that the reader is afforded the insight that is gained from analytical solution as well as the numerical solution that must often be used in practice. Analytical we used the techniques of separation of variables. It is worthwhile to mention here that, analytical solution is not always possible to obtain; indeed, in many instants they are very cumber some and difficult to use. In that case a numerical technique is more appropriate. Among numerical techniques finite difference schema is used. In both approach we found a solution which agrees up to one decimal place.
Abstract: The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between materia...
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