scholarly journals Solving Nonlinear Thermal Problems of Friction by Using Method of Lines

2015 ◽  
Vol 9 (1) ◽  
pp. 33-37
Author(s):  
Ewa Och
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Heat Conduction Problem ◽  
Method Of Lines ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Partial Linearization ◽  
The Method Of Lines ◽  
Third Stage ◽  
Independent Variable

Abstract One-dimensional heat conduction problem of friction for two bodies (half spaces) made of thermosensitive materials was considered. Solution to the nonlinear boundary-value heat conduction problem was obtained in three stages. At the first stage a partial linearization of the problem was performed by using Kirchhoff transform. Next, the obtained boundary-values problem by using the method of lines was brought to a system of nonlinear ordinary differential equations, relatively to Kirchhoff’s function values in the nodes of the grid on the spatial variable, where time is an independent variable. At the third stage, by using the Adams's method from DIFSUB package, a numerical solution was found to the above-mentioned differential equations. A comparative analysis was conducted (Och, 2014) using the results obtained with the proposed method and the method of successive approximations.

Effortless Application of the Method of Lines for the Inverse Estimation of Temperatures in a Large Slab With Two Different Surface Heating Waveforms

2008 ◽  
Vol 131 (2) ◽  
Author(s):  
Antonio Campo ◽  
John Ho
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Method Of Lines ◽  
Time History ◽  
Numerical Differentiation ◽  
Heat Fluxes ◽  
Exposed Surface ◽  
The Method Of Lines ◽  
Temperature Time

The boundary inverse heat conduction problem (BIHCP) deals with the determination of the surface heat flux or the surface temperature from measured transient temperatures inside a conducting body where the initial temperature is known. This work addresses a BIHCP related to the spatiotemporal heat conduction in a large slab when a time-variable heat flux is prescribed at an exposed surface and the other surface is thermally insulated. Two different heating waveforms are studied: a constant heat flux and a time-dependent triangular heat flux. The numerical temperature-time history at the insulated surface of the large slab provides the “temperature-time measurement” with one temperature sensor. Framed in the theory of the method of lines (MOL) first and employing rudimentary concepts of numerical differentiation later, the main objective of this paper is to develop a simple computational methodology to estimate the temporal evolution of temperature at the exposed surface of the large slab receiving the two distinct heat fluxes. In the end, it is confirmed that excellent predictions of the surface temperatures versus time are achievable for the two cases tested while employing the smallest possible system of two heat conduction differential equations of first-order.

The Numerical Method of Lines facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces

2020 ◽  
pp. 030641902091042
Author(s):  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Method Of Lines ◽  
Adjoint System ◽  
One Dimensional ◽  
First Order ◽  
The Method Of Lines ◽  
Unsteady Heat Conduction

The present study on engineering education addresses the Method of Lines and its variant the Numerical Method of Lines as a reliable avenue for the numerical analysis of one-dimensional unsteady heat conduction in walls, cylinders, and spheres involving surface convection interaction with a nearby fluid. The Method of Lines transforms the one-dimensional unsteady heat conduction equation in the spatial and time variables x, t into an adjoint system of first-order ordinary differential equations in the time variable t. Subsequently, the adjoint system of first-order ordinary differential equations is channeled through the Numerical Method of Lines and the powerful fourth-order Runge–Kutta algorithm. The numerical solution of the adjoint system of first-order ordinary differential equations can be carried out by heat transfer students employing appropriate routines embedded in the computer codes Maple, Mathematica, Matlab, and Polymath. For comparison, the baseline solutions used are the exact, analytical temperature distributions that are available in the heat conduction literature.

scholarly journals Random Fuzzy Differential Equations with Impulses

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Ho Vu
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Picard Method

We consider the random fuzzy differential equations (RFDEs) with impulses. Using Picard method of successive approximations, we shall prove the existence and uniqueness of solutions to RFDEs with impulses under suitable conditions. Some of the properties of solution of RFDEs with impulses are studied. Finally, an example is presented to illustrate the results.

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