Adv. Studies Theor. Phys., Vol. 8, 2014, no. 10, 463 - 469

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/astp.2014.4443

Analytical Solution to an Attic, Basement, and

Insulated Main Floor Home Heating Systems

Iyad Suwan

Department of Mathematics and Statistics

Arab American University Jenin, Jenin, Palestine

Copyright © 2014 Iyad Suwan. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract

In this paper, a general system of ordinary differential equations that models the

home heating system of a typical home with attic, basement, and main living area is

solved. Newton’s Law of Cooling is used to analyze the temperature changes in the

system. Under certain conditions on the cooling coefficients, the model is reduced to

a third order ordinary differential equation. Solving this equation gives the solution

of the system.

Keywords: Linear Systems of ODEs, High Order, Home Heating, Newton’s Cooling

System.

Introduction

Developing new and efficient methods for solving systems of linear ordinary

differential equations (ODEs) is still under consideration of many researches in

464 Iyad Suwan

applied mathematics, physics, engineering and many branches of science. In

additional to their importance in representing many physical models, linear systems

can be used to facilitate solving nonlinear ones when they share the same structure of

symmetry [25]. Therefore, many analytical and numerical approaches are developed

in the last and in the present centuries to provide solutions of linear systems[1-27].

One of the most important physical applications of those systems is modeling the

temperature distribution of a typical home with attic, basement, and main floor.

In this paper, a general condition on the cooling coefficients is proposed in order to

convert the system of ODEs into a single equation. The resulting equation can be

solved by considering the roots of the characteristic equation when it is

homogeneous. When the ODE is non-homogeneous; the method of variations of

parameters can be used.

The home system considered in this paper is an insulated main floor (living area),

and un-insulated attic and basement. The functions ( ) ( ) ( ) represent the

attic temperature, the main floor temperature, and the basement temperature

respectively at any time According to Newton’s Cooling Law, the following

system of ODEs models the considered home heating system:

( ) ( ( )) ( ( ) ( ))

( ) ( ( ) ( )) ( ( )) ( ( ) ( ))

( ) ( ( ) ( )) ( ( )) (1)

Where ( ) ( ) ( ) is the provided temperature to the main

floor per unit time, and are the cooling coefficients determined by the

Newton’s Cooling Law and depend on the system. In this paper, the cooling

coefficients will be considered arbitrary real numbers.

In the next section, System (1) is analyzed and converted to a general third order

ODE. Conclusions and future perspectives are presented in the last section.

Analysis of the Method and Main Result

The system (1) can be written in the form:

Analytical solution 465

( )

( )

( ) (2)

If we let , , , ,

, and , then (2) is

(3)

From (3),

(4)

and

(5)

Using (4). Equation (5) becomes

(

)

(6)

Substituting for and , (6) is written as

( ) ( ) +

(7)

Rearranging terms in (7) leads to

( ) (

)

+

(8)

466 Iyad Suwan

Now, if then (8) becomes

( )

(9)

But from (3),

( ( ) ) (10)

So (9) can be written as

(

)( )

(11)

Finally, rearranging terms in (11) leads to

(

) ( (

))

(

) (12)

Equation (12) is a third order non-homogenous ODE, solving it for gives easy

finding of and in (1).

Conclusions and Future Perspectives

A typical heating system in a home with insulated main floor, and un-insulated

basement and attic are considered in this paper. Using Newton’s cooling law, the

heating system is modeled by a system of linear ODEs. A certain condition is put on

the cooling coefficients in order to convert the system to one third order ODE.

Solving this equation for the temperature in the basement at any time can be used

to find temperature at the main floor and the attic. As a future work, the conditions

on the cooling coefficients will be under consideration in order to give a general

technique for any home heating systems.

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Analytical solution 467

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Received: April 3, 2014