Reach Us
+44-1474-556909

^{1}Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria

^{2}Department of Mathematics, Federal University Dutse, Jigawa, Nigeria

- *Corresponding Author:
- Sunday Babuba

Department of Mathematics

Federal University Dutse

Jigawa, Nigeria

**Tel:**+234 807 079 3965

**E-mail:**[email protected]

**Received Date:** June 25, 2017; **Accepted Date:** July 22, 2017; **Published Date:** July 28, 2017

**Citation: **Odekunle MR, Babuba S (2017) A New Numerical Method of Estimates
of Temperatures along a Thick Steel Slab and Concentrations of Alcohol along
a Hollow Tube. J Appl Computat Math 6: 355. doi: 10.4172/2168-9679.1000355

**Copyright:** © 2017 Odekunle MR, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

A new continuous numerical method based on the approximation of polynomials is here proposed for solving the equation arising from heat transfer along a thick steel slab and a hollow tube subject to initial and boundary conditions. The method results from discretization of the heat equation which leads to the production of a system of algebraic equations. By solving the system of algebraic equations we obtain the problem approximate solutions.

Polynomials; Interpolation; Multistep collocation; Heat conduction

The development of numerical techniques for solving heat conduction equation in science and engineering subject to initial and boundary conditions is a subject of considerable interest. In this paper, we develop a new continuous numerical method which is based on interpolation and collocation at some point along the coordinates (Odekunle, 2008). To do this we let U(x,t) represents the temperature at any point in the slab and the tube. Heat is flowing from one end to another under the influence of the temperature gradient ∂U/∂x. To make a balance of the rate of heat flow in and out of the media, we consider *R* for thermal conductivity of the steel, *C* the heat capacity which we assume constants, and *ρ* the density and *D* the thermal diffusivity of alcohol [1]. Heat flow in the slab is given by

(1.0)

Heat flow through the tube is also given by

(1.1)

Where *A* and *B* are the cross sections of the slab and the tube respectively.

The new method strives to provide solutions to the heat flow eqns. (1.0) and (1.1).

To set up the solution method we select an integer N such that N>0. We subdivide the interval 0 ≤ x ≤ x into N equal subintervals with mesh points along space axis given by , where Nh=X, similarly, we reverse the roles of x and t and we select another integer M such that M > 0. We also subdivide the interval 0 ≤ t ≤ T into M equal subintervals with mesh points along the time axis given by where MK=T and h,k are the mesh sizes along space and time axes, respectively [2]. Here, we seek for the approximate solution to U(x,t) of the form

(2.0)

Over h > 0, k > 0 mesh sizes, such that

is the sum of interpolation points along the space and time coordinates. That is *ρ* = *g* + *b*, where *g* is the number of interpolation points along the space axis and *b* the number of interpolation points along time coordinate. The basis function is the Taylor’s polynomials which is known, ar are the constants to be determined. There will be flexibility in the choice of the basis function as may be desired for specific application. For this work, we consider the Taylor’s polynomials [3-7].

The interpolation values are assumed to have been determined from previous steps, while the method seeks to obtain (Odekunle, 2008). Applying the above interpolation conditions on eqn. (2.0) we obtain,

(2.1)

We let arbitrarily and k=0, then by Crammer’s rule, eqn. (2.1) becomes

(2.2)

and

Where and W^{-1} exists (Odekunle, 2008]). Hence, by equation (2.2) we obtain

(2.3)

The vector is now determined in terms of known parameters in . If is the row of then

(2.4)

Eqn. (2.4) determines the values [8-17]. Let us take first and second derivatives of eqn. (2.0) with respect tox,

(2.5)

Substituting eqn. (2.4) into eqn. (2.5), we obtain

(2.6)

We reverse the roles of *x* and *t* in eqn. (2.1) and we arbitrarily set and h=0, by Crammer‘s rule eqn. (2.1) becomes

(2.7)

Where , and Y^{-1} exists (Odekunle, 2008). Hence,

from eqn. (2.7) we obtain [4]

(2.8)

The vector is now determined in terms of known parameters in *L E*. [21-23]. If

(2.9)

Also, eqn. (2.9) determines the values of *a _{r}*. Taking the first derivatives of eqn. (2.0) with respect to t, we obtain

(2.10)

Substituting eqn. (2.9) in eqn. (2.10) we have

(2.11)

But by eqn. (1.0) or (1.1) it is obvious that eqn. (2.11) is equal to eqn. (2.6), therefore,

(2.12)

Collocating eqn. (2.12) at x=x_{i} and t=t_{j} we obtain a new numerical scheme that solves eqns. (1.0) and (1.1) explicitly.

In this section we give some numerical examples to compute approximate solutions for equations (1.0) and (1.1) by the method discussed in this paper [5]. This is in order to test the numerical accuracy of the new method. To achieve this, we truncate the Taylor’s polynomial after second degree and use it as the basis function in the computation. The resultant scheme is used to solve the following two problems.

Example 1 (Eyaya, 2010)

Given a 2 cm thick steel slab, solve for the temperatures as a function of x and t at t=2.062 seconds if the initial temperatures are given by the relation [24-26]. where *k* for steel is 0.13 cal/sec °cm, c=0.11 cal/g°C and p=7.8 g/cm^{3}.

Solution

By simplification eqn. (1.0) becomes . To solve this equation we take Δx=0.25 cm, then we find Δ*t* by the relation , Δ*t*=0.825 sec.We let β=4, α=64 arbitrarily which implies that . Taking two interpolation points along space coordinates and one along time implies that g=2,b=1,p=3 and for we obtain , then the calculated temperatures are tabulated as shown in **Table 1** [27].

T | x=0 |
x=0.25 |
X=0.50 |
x=0.75 |
x=1.00 |
x=1.25 |
---|---|---|---|---|---|---|

0.0 | 0.0 | 38.27 | 70.71 | 92.39 | 100 | 92.39 |

0.825 | 0.0 | 37.54 | 69.37 | 90.63 | 98.10 | 90.63 |

1.65 | 0.0 | 36.83 | 68.05 | 88.91 | 96.23 | 88.91 |

2.475 | 0.0 | 36.13 | 66.76 | 87.23 | 94.40 | 87.23 |

3.3 | 0.0 | 35.45 | 65.49 | 85.59 | 92.61 | 85.59 |

4.125 | 0.0 | 34.77 | 64.24 | 83.94 | 90.85 | 83.94 |

**Table 1:** Calculated temperatures.

Example 2 (Eyaya, 2010)

A hollow tube 25 cm long is initially filled with air containing 2% of ethyl alcohol vapors. At the bottom of the tube is a pool of alcohol which evaporates in to the stagnant gas above [6]. (Heat transfers to the alcohol from the surroundings to maintain a constant temperature of, 30°C at which temperature the vapor pressure is 0.1 atm.). At the upper end of the tube, the alcohol vapors dissipate to the outside air, so the concentration is essentially zero. Considering only the effects of molecular diffusion, determine the concentration of alcohol as a function of time and distance measured from the top of the tube.

Solution

Molecular diffusion follows the law where D is the diffusion coefficient, with units in cm2/sec. (This is the same as for the ratio k/cp which is often termed thermal diffusivity). For ethyl alcohol D=0.111 cm2/sec at 30°c, and the vapor pressure is such that 10 volume percent alcohol in air is present at the surface [7]. The initial condition is c(x,0)=3.0 and the boundary conditions are given by c(0,t)=0, c(25,t)=15. Since the length of the tube is 25 cm, we take Δ*x*=5 cm using the maximum value permitted for Δ*t* yields

If we take β=4 , α=3 arbitrarily then and . Taking two interpolation points along space coordinates and one along time implies that g=2, b=1, implies that p=3 and for we obtain then the calculated concentrations of alcohol are tabulated as shown in **Table 2**.

t | x=0 |
x=5 |
x=10 |
x=15 |
x=20 |
x=25 |
---|---|---|---|---|---|---|

0 | 0 | 3.00 | 3.00 | 3.00 | 3.00 | 15 |

21.11 | 0 | 2.63 | 3.00 | 3.00 | 4.50 | 15 |

42.22 | 0 | 2.35 | 2.95 | 3.19 | 5.63 | 15 |

63.33 | 0 | 2.13 | 2.91 | 3.47 | 6.50 | 15 |

84.44 | 0 | 1.96 | 2.88 | 3.78 | 7.18 | 15 |

105.55 | 0 | 1.83 | 2.88 | 4.09 | 7.73 | 15 |

126.66 | 0 | 1.73 | 2.90 | 4.39 | 8.18 | 15 |

147.77 | 0 | 1.66 | 2.94 | 4.68 | 8.56 | 15 |

168.88 | 0 | 1.61 | 2.99 | 4.95 | 8.88 | 15 |

**Table 2:** Concentrations of alcohol.

- Adam A, David R (2002) One dimensional heat equation.
- Awoyemi DO (2002) An Algorithmic collocation approach for direct solution of special fourth-order initial value problems of ordinary differential equations. Journal of the Nigerian Association of Mathematical Physics 6: 271-284.
- Awoyemi DO (2003) Ap-stable linear multistep method for solving general third order Ordinary differential equations. Int J Computer Math 80: 987-993.
- Bao W, Jaksch P, Markowich PA (2003) Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J Compt Phys 187: 318-342.
- Benner P, Mena H (2004) BDF methods for large scale differential Riccati equations in proc. of mathematical theory of network and systems, MTNS.
- Bensoussan A, Da Prato G, Delfour M, Mitter S (2007) Representation and control of infinite dimensional systems. Sys Cont Found Appl 2:576
- Biazar J, Ebrahimi H (2005) An approximation to the solution of hyperbolic equation by a domain decomposition method and comparison with characteristics method. Appl Math Comput 163: 633-638
- Brown PLT (1979) A transient heat conduction problem. AICHE Journal 16: 207-215.
- Chawla MM, Katti CP (1979) Finite difference methods for two-point boundary value problems involving high-order differential equations. BIT. 19: 27-33.
- Cook RD (1974) Concepts and Application of Finite Element Analysis.NY: Wiley Eastern Limited.
- Crandall SH (1955) An optimum implicit recurrence formula for the heat conduction equation. JACM 13:318-320.
- Crane RL, Klopfenstein RW (1965) A predictor-corrector algorithm with increased range of absolute stability. JACM 12: 227-241.
- Crank J, Nicolson P (1947) A practical method for numerical evaluation of solutions of partial differential equations of heat conduction type. Proc Camb Phil Soc 6: 32-50.
- Dahlquist G, Bjorck A (1974) Numerical methods. NY: Prentice Hall.
- Dehghan M (2003) Numerical solution of a parabolic equation with non-local boundary specification. Appl Math Comput 145: 185-194.
- Dieci L (1992) Numerical analysis. SIAM Journal 29: 781-815.
- Douglas J (1961) A Survey of Numerical Methods for Parabolic Differential Equations in advances in computer II. Academic press.
- D’Yakonov Ye G (1963) On the application of disintegrating difference operators. Z Vycist Mat I Mat Fiz. 3: 385-388.
- Eyaya BE (2010) Computation of the matrix exponential with application to linear parabolic PDEs.
- Fox L (1962) Numerical Solution of Ordinary and Partial Differential Equation.
- Penzl T (2000) Matrix analysis. SIAM J 21: 1401-1418.
- Pierre J (2008) Numerical solution of the dirichlet problem for elliptic parabolic equations. SIAM J. Soc. Indust Appl Math 6: 458-466.
- Richard LB Albert C (1981) Numerical analysis. Berlin: Prindle, Weber and Schmidt inc.
- Richard L, Burde J, Douglas F (2001) Numerical analysis. (7th edn.), Berlin: Thomson Learning Academic Resource Center.
- Saumaya B, Neela N, Amiya YY (2012) Semi discrete Galerkin method for equations of Motion arising in Kelvin-Voitght model of visco-elastic fluid flow. Journal of Pure and Applied Science 3: 321- 343.
- Yildiz B, Subasi M (2001) On the optimal control problem for linear Schrodinger equation. Appl Math and Comput121: 373-381.
- Zheyin HR, Qiang X (2012) An approximation of incompressible miscible displacement in porous media by mixed finite elements and symmetric finite volume element method of characteristics. App Mathe Compu Else 143: 654-672.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebraic Geometry
- Analytical Geometry
- Applied Mathematics
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Binary and Non-normal Continuous Data
- Binomial Regression
- Biometrics
- Biostatistics methods
- Clinical Trail
- Complex Analysis
- Computational Model
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Differential Equations
- Differential Transform Method
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Hamilton Mechanics
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Physical Mathematics
- Quantum Mechanics
- Quantum electrodynamics
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Riemannian Geometry
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topology
- mirror symmetry
- vector bundle

- Total views:
**768** - [From(publication date):

July-2017 - Dec 06, 2019] - Breakdown by view type
- HTML page views :
**678** - PDF downloads :
**90**

**Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals**

International Conferences 2019-20