ISSN-1995-0756 EISSN-1998-1066 Journal home page: http://www.aensiweb.com/aeb.html

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AENSI Journals

Advances in Environmental Biology

ISSN-1995-0756 EISSN-1998-1066

Journal home page: http://www.aensiweb.com/aeb.html

Corresponding Author: E. Hashemizadeh, Young Researchers and Elite Club, Karaj Branch, Islamic Azad University, Karaj, Iran.

E-mail: [email protected]

A Numerical Approach for the Solution of Nonlinear Boundary Value Problems Arising in Biology Via Shifted Jacobi Operational Matrix

1E. Hashemizadeh and ²F. Mahmoodi

1Young Researchers and Elite Club, Karaj Branch, Islamic Azad University, Karaj, Iran.

2Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

A R T I C L E I N F O A B S T R A C T

Article history:

Received 28 February 2014 Received in revised form 19 April 2014

Accepted 23 April 2014 Available online 25 May 2014

Key words:

Jacobi polynomials, operational

matrix of

derivative,nonlineardifferential equations, biology, physiology.

The objective of this paper is to apply the shifted Jacobi polynomials on some nonlinear ordinary differential equations that arise in various biology problems. Some numerical illustrations in biology are given; also we compared present method results by some existed method results.

To Cite This Article: E. Hashemizadeh and F. Mahmoodi., A Numerical Approach for the Solution of Nonlinear Boundary Value Problems Arising in biology Via Shifted Jacobi Operational Matrix. Adv. Environ. Biol., 8(5), 1415-1419, 2014

INTRODUCTION

Analytical methods commonly used to solve nonlinear differential equations are very restricted and numerical techniques involving discretization of the other hand gives rise to rounding off errors. These are indispensable tools for modeling many physiology problems such as study of steady state oxygen-diffusion in a cell with Michaelis-Menten uptake kinetics [1-2] spring mass system [3] and bending of beams [4]. These equations are also useful in study of the distribution of heat sources in the human head [5-6] and tumor growth [7-11].

The aim of this paper is to introduce a new method for the numerical solution of the following class of singular boundary value problems

𝑦^′′ 𝑥 + 𝑎 +^𝑚

𝑥 𝑦^′= 𝑓 𝑥, 𝑦 , 0 ≤ 𝑥 ≤ 1, (1) 𝛼₁𝑦 0 + 𝛽₁𝑦^′ 0 = 𝛾₁, (2) 𝛼₂𝑦 1 + 𝛽₂𝑦^′ 1 = 𝛾₂, (3)

Which arising in biology and physiology problems. We assume that 𝑓 𝑥, 𝑦 is continuous, ^𝜕𝑓

𝜕𝑦exists and is continuous and also^𝜕𝑓

𝜕𝑦≥ 0 , ∀𝑥; 0 ≤ 𝑥 ≤ 1.Theboundary value problem (1)-(3) with 𝑚 = 0,1,2 and 𝛼 = 0 arise in the study of various tumor growth problems, see[12-13], with linear 𝑓 𝑥, 𝑦 and with nonlinear 𝑓 𝑥, 𝑦 of the form

𝑓 𝑥, 𝑦 ≡ 𝑓 𝑦 = ^𝑛𝑦

𝑦+𝜇 , 𝑛 > 0, 𝜇 > 0. (4) A mathematical model of tumor growth is a mathematical expression of the dependence of tumor size on time.

And when𝑚 = 2, 𝛼 = 0 in modeling of conduction in human head, see[14-15], with

𝑓 𝑥, 𝑦 ≡ 𝑓 𝑦 = −δe^−yσ , σ > 0, 𝛿 > 0. (5) Existence-uniqueness results for such problems have been established by several researchers[16-22].

In this paper by use of shifted Jacobi polynomials as basis and operational matrix of derivatives [23], of them we convert these kinds of equations to nonlinear algebraic equations. The advantage of this method analogy to other existed method for these problems is its trusty and simply in implementation, we compared our results with some existed results to prove this claim.

This paper is organized as follows: Section 2 represents preliminaries; in this section we introduced shiftedJacobi polynomials, and some properties of them, especiallythe operational matrix of derivative. In

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Section 3 we implemented them on physiology problems. In Section 4, some applied models in physiology are discussed to show the efficiency and accuracy of the proposed method, the results obtained are compared by the known results. Finally, Section 5 includes a conclusion for the paper.

2. Shifted Jacobi polynomials and their operational matrix of derivative:

2.1. Shifted Jacobi Polynomials:

The well-known Jacobi polynomials are defined on the interval −1,1 and can be generated with the aid of the following recurrence formula:

P_i^α,β t = α + β + 2i − 1 α²− β²+ t α + β + 2i α + β + 2i − 2

2i α + β + i α + β + 2i − 2 P_i−1^α,β t

− α + i − 1 β + i − 1 α + β + 2i

i α + β + i α + β + 2i − 2 P_i−2^α,β t , i = 2,3, ⋯ Where

P₀^α,β t = 1 andP₁^α,β t =^{𝛼+𝛽 +2}

2 𝑡 +^𝛼−𝛽

2 .

In order to use these polynomials on the interval 𝑥 𝜖 0, 𝐿 Doha and et al in [23]derived the so-called Shifted Jacobi polynomials by introducing the change of variable𝑡 =^2𝑥

𝐿 − 1. Let the ShiftedJacobi polynomials P_L,i^α,β ^2𝑥

𝐿 − 1 be denoted byP_L,i^α,β x . Then P_L,i^α,β x can be generated form:

P_L,i^α,β x = α + β + 2i − 1 α²− β²+ ^2𝑥

𝐿 − 1 α + β + 2i α + β + 2i − 2

2i α + β + i α + β + 2i − 2 P_L,i−1^α,β x

− α + i − 1 β + i − 1 α + β + 2i

i α + β + i α + β + 2i − 2 P_L,i−2^α,β x , i = 2,3, ⋯ Where

P_L,0^α,β 0 = 1andP_L,1^α,β 1 =^{𝛼+𝛽 +2}

2 2𝑥

𝐿 − 1 +^𝛼−𝛽

2 .

The analytic form of the Shifted Jacobi polynomialsP_L,i^α,β x of degree 𝑖 isgiven by P_L,i^α,β x = −1 ^𝑖−𝑘

𝑖

𝑘=0

Γ 𝑖 + 𝛽 + 1 Γ 𝑖 + 𝑘 + 𝛼 + 𝛽 + 1

Γ 𝑘 + 𝛽 + 1 Γ 𝑖 + 𝛼 + 𝛽 + 1 𝑖 − 𝑘 ! 𝑘! 𝐿^𝑘𝑋^𝑘. Where

P_L,i^α,β 0 = −1 𝑖 Γ 𝑖+𝛽 +1

Γ 𝛽 +1 𝑖! , P_L,i^α,β L =^{Γ(𝐼+𝛼+1)}

Γ 𝛼+1 𝑖!. 2.2. Function Approximation:

The orthogonally condition of Shifted Jacobi polynomials is P_L,j^α,β x P_L,k^α,β x 𝑊_𝐿^{𝛼 ,𝛽}

𝐿

0

𝑥 𝑑𝑥 = ℎ_𝑘,

Where 𝑊_𝐿^{𝛼 ,𝛽} 𝑥 = 𝑥^𝛽 𝐿 − 𝑥 ^𝛼 and ℎ𝑘=

𝐿^{𝛼 +𝛽 +1}Γ 𝐾+𝛼+1 Γ 𝐾+𝛽 +1

2𝐾+𝛼 +𝛽 +1 𝐾!Γ 𝐾+𝛽 +𝛼 +1 , 𝑖 = 𝑗, 0 𝑖 ≠ 𝑗.

Let 𝑢(𝑥) be a polynomial of degree n, thenit may be expressed in terms of Shifted Jacobi polynomials as 𝑢 𝑥 = 𝑁 𝑐𝑗

𝑗 =0 𝑃_𝐿,𝑗^𝛼,𝛽 𝑥 = 𝐶^𝑇Φ 𝑥 , (6) Where the coefficients 𝑐𝑗 are given by

𝑐_𝑗 = 1 ℎ𝑗

𝑊_𝐿^{𝛼 ,𝛽} 𝑥 𝑢(𝑥)

𝐿

0

P_L,j^α,β x 𝑑𝑥. 𝑗 = 0,1, ⋯

If the Shifted Jacobi coefficient vector C and the Shifted Jacobivector Φ 𝑥 are written as

𝐶^𝑇= 𝑐₀, 𝑐₁, … , 𝑐_𝑁 , (7) and

Φ 𝑥 = P_L,0^α,β x , P_L,1^α,β x , … , P_L,N^α,β x ^𝑇. (8)

2.3.Operational Matrix of Derivative:

The derivative of the vector Φ 𝑥 can be expressed by

𝑑Φ 𝑥

𝑑𝑥 = 𝑫⁽¹⁾ Φ 𝑥 , (9) Where 𝑫⁽¹⁾ is the 𝑁 + 1 × 𝑁 + 1 Operational matrix of derivative given by

𝑫⁽¹⁾= 𝑑𝑖𝑗 = 𝐶1 𝑖, 𝑗 𝑖 > 𝑗, 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,

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Where

𝐶₁ 𝑖, 𝑗 =^𝐿^{𝛼 +𝛽} 𝑖+𝛼+𝛽+1 𝑖+𝛼 +𝛽 +2 _𝑗 𝑗 +𝛼 +2 _{𝑖−𝑗 −1}Γ 𝑗 +𝛼+𝛽 +1

𝑖−𝑗 −1 !Γ 2𝑗 +𝛼 +𝛽 +1 × 𝐹₃ ₂ −𝑖+1+𝑗 , 𝑖+𝑗 +𝛼+𝛽 +2, 𝑗 +𝛼+1 𝑗 +𝛼 +2, 2𝑗 +𝛼 +𝛽 +2 ; 1 .

(for the proof, see[24], and for the general definition of a generalized hyper geometric series and special3𝐹2 , see [25], p. 41 and pp. 103-104, respectively).

For example, for even N we have

 

    

   

     

1

1 1

1

1 1

1 1 1

0 0 0 0

1, 0 0 0 0

2, 0 2,1 0 0

3, 0 3,1

0 0

, 0 ,1 , 1 0

C

C C

C N C N C N N

 

 

  

 

  

 



  

 

 D

. By using (9), it is clear that

𝑑^𝑛Φ(𝑥)

𝑑𝑥^𝑛 = 𝑫¹ ^𝑛𝛷 𝑥 . (10) Where 𝑛 𝜖 𝑁and the superscript in𝑫¹, denotes matrix powers. Thus

𝑫¹= 𝑫¹ ^𝑛, 𝑛 = 1,2, ⋯.

3. Implementation of Shifted Jacobi Polynomials Method on biology Problems:

In this section we use shifted Jacobi vector and its operational matrix of derivative to solve nonlinear singular boundary value problem of the form Eq. (1) with mixed conditions (2), and (3).

Form Eq. (6) we can approximate our unknown as

𝑦 𝑥 = 𝐶^𝑇𝚽 𝑥 , (11) Where 𝚽 𝑥 and C are defined in Eqs. (7) and (8). By using Eqs. (9) and (10) we have

𝑦^′ 𝑥 = 𝐶^𝑇𝚽^′ 𝑥 = 𝐶^𝑇𝑫⁽¹⁾𝚽 𝑥 , (12) and

𝑦^′′ 𝑥 = 𝐶^𝑇𝚽^′′ 𝑥 = 𝐶^𝑇 𝑫⁽¹⁾ ²𝚽 𝑥 . (13) By substituting Eqs.(11), (12) and (13) in Eq. (1) we have

𝐶^𝑇 𝑫⁽¹⁾ ²𝚽 𝑥 + 𝑎 +^𝑚

𝑥 𝐶^𝑇𝑫¹𝚽 𝑥 = 𝑓 𝑥, 𝐶^𝑇𝚽 𝑥 . (14)

Alsoby using Eqs. (2), (3), (11) and (12) we have

𝛼₁𝐶^𝑇𝚽 0 + 𝛽₁𝐶^𝑇𝑫⁽¹⁾𝚽 0 = 𝛾₁, (15) 𝛼2𝐶^𝑇𝚽 1 + 𝛽2𝐶^𝑇𝑫⁽¹⁾𝚽 1 = 𝛾2. (16) Eqs. (15) and (16) give two linear equations. Since the total unknowns for vector C in Eq. (11) is(𝑁 + 1), we collocate Eq. (14) in (𝑁 − 1) points 𝑥𝑖 in the interval 0,1 as

𝑥_𝑝= ^2𝑝−1

2(𝑛+1) , 𝑝 = 1,2, … , 𝑛 − 1 . (17) Then we will have

𝐶^𝑇 𝑫⁽¹⁾ ²𝚽 𝑥_𝑖 + 𝑎 +^𝑚

𝑥_𝑖 𝐶^𝑇𝑫¹𝚽 𝑥 = 𝑓 𝑥, 𝐶^𝑇𝚽 𝑥_𝑖 . (18) For 𝑖 = 1,2, … , 𝑁 − 1. Now the resulting Eqs. (15), (16) and (18) generate a system of (𝑁 + 1) nonlinear equations which can be solved using Newton’s iterative method [26-27].We used the Mathematica 8 software to solve this nonlinear system.

4. Some applied models in biology:

4.1.Example 1: Oxygen diffusion:

Consider the following oxygen diffusion problem 𝑦^′′ 𝑥 +2

𝑥𝑦^′ 𝑥 = 0.76129𝑦 𝑦 + 0.03119, 𝑦^′ 0 = 0, 5𝑦 1 + 𝑦^′ 1 = 5.

As this problem is a real world problem we haven’t its exact answer, because of this we compare different numerical method answers for this example [28-31] that are presented in Table 1. For the proposed method we used three choices for 𝛼 and 𝛽, for 𝑁 = 13.

4.2. Example 2: Heat conduction:

Consider this problem that is coincide by heat conduction model of the human head

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𝑦^′′ 𝑥 +2

𝑥𝑦^′ 𝑥 = −𝑒^−𝑦, 𝑦^′ 0 = 0, 𝑦 1 + 𝑦^′ 1 = 0.

Table 2 shows the different methods results for this example [28,29,32,33] analogy to present method results.

5. Conclusion:

This paper present a numerical method, based on shifted Jacobipolynomials for the numerical solution of a class of singular boundary value problems arising in biology. By use of shifted Jacobi polynomials as basis and use operational matrix of derivative of these functions we convert such problems to a nonlinear system that can be solved by Newton’s. The implementation of current approach is very simple, and we can execute this method on a computer speedy. The numerical examples that have been presented in the paper and the compared results support our claim.

Table 1: Approximate solutions for Example 1.

Present method with 𝛼 = 0, 𝛽 = 0

𝑁 = 13

Present method with 𝛼 = 0.5, 𝛽 = 0.5

𝑁 = 13

Method in [28]

With 𝑛 = 14

Method in [29]

With 𝑛 = 15

Method in [30]

With 𝑛 = 20

Method in [31]

With 𝑛 = 60

0.0 0.82848329035991 0.82848329035986 0.82848329035987 0.82848329035981 0.82848329035968 0.82848329481355 0.82848327295802 0.1 0.82970609243403 0.82970609243398 0.82970609243399 0.82970609243393 0.82970609243380 0.82970609688790 0.82970607521884 0.2 0.83337473359123 0.83337473359118 0.83337473359119 0.83337473359113 0.83337473359100 0.83337473804308 0.83337471691089 0.3 0.83948991395393 0.83948991395388 0.83948991395389 0.83948991395383 0.83948991395370 0.83948991833986 0.83948989814383 0.4 0.84805278499629 0.84805278499624 0.84805278499625 0.84805278499619 0.84805278499606 0.84805278876051 0.84805277036165 0.5 0.85906492716946 0.85906492716941 0.85906492716942 0.85906492716936 0.85906492716923 0.85906492753032 0.85906491397434 0.6 0.87252831995851 0.87252831995846 0.87252831995848 0.87252831995841 0.87252831995828 0.87252831569855 0.87252830841853 0.7 0.88844530562342 0.88844530562337 0.88844530562338 0.88844530562332 0.88844530562319 0.88844529949702 0.88844529589927 0.8 0.90681854806703 0.90681854806698 0.90681854806699 0.90681854806693 0.90681854806680 0.90681854179965 0.90681854026297 0.9 0.92765098836581 0.92765098836576 0.92765098836577 0.92765098836571 0.92765098836558 0.92765098305256 0.92765098252660 1.0 0.95094579849668 0.95094579849663 0.95094579849664 0.95094579849659 0.95094579849648 0.95094579480523 0.95094579461056

Table 2: Approximate solutions for Example 2.

𝑁 = 13

Present method with 𝛼 = 0.5, 𝛽 = 0.5

𝑁 = 13

Method in [28]

With 𝑛 = 14

Method in [29]

With 𝑛 = 15

Method in [32]

With Forth-order

Method in [33]

0.0 0.36751681512888 0.36751681512889 0.36751681512876 0.3675168151 0.3675168151 0.3675181074 0.3675169710 0.1 0.36636232924027 0.36636232924028 0.36636232924014 0.3663623292 0.3663623292 0.3663637561 0.3663623697 0.2 0.36289406611898 0.36289406611899 0.36289406611886 0.3628940661 0.3628940661 0.3628959378 0.3628941066 0.3 0.35709754572096 0.35709754572097 0.35709754572084 0.3570975457 0.3570975457 0.3570991429 0.3570975842 0.4 0.34894842062160 0.34894842062161 0.34894842062147 0.3489484206 0.3489484206 0.3489499903 0.3489484612 0.5 0.33841214875014 0.33841214875015 0.33841214875001 0.3384121487 0.3384121487 0.3384136581 0.3384121893 0.6 0.32544352243403 0.32544352243404 0.3254435224339 0.3254435224 0.3254435224 0.3254450019 0.3254435631 0.7 0.30998604023188 0.30998604023189 0.30998604023174 0.3099860402 0.3099860402 0.3099878567 0.3099860810 0.8 0.29197110305610 0.29197110305612 0.29197110305597 0.2919711030 0.2919711030 0.2919789654 0.2919711440 0.9 0.27131701015804 0.27131701015805 0.27131701015790 0.2713170101 0.2713170101 0.2713185637 0.2713170512 1.0 0.24792772331700 0.24792772331701 0.24792772331686 0.2479277233 0.2479277233 0.2479292837 0.2479277646

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