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Research Article
 

Fourier Transform Solution of the Semi-linear Parabolic Equation



E.O. Oghre and B.I. Olajuwon
 
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ABSTRACT

The semi-linear heat equation models many physical processes, which include time-dependent, irreversible processes such as that of conduction, chemical reactions and biological flow problems. We have considered the one dimensional semi-linear heat equation and a piecewise continuous integrable function f(x, t). Using a Fourier cosine transform and a backward inverse operation we determined the solution to the problem.

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  How to cite this article:

E.O. Oghre and B.I. Olajuwon, 2005. Fourier Transform Solution of the Semi-linear Parabolic Equation. Journal of Applied Sciences, 5: 492-495.

DOI: 10.3923/jas.2005.492.495

URL: https://scialert.net/abstract/?doi=jas.2005.492.495

INTRODUCTION

We consider the problem

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

This equation models many physical processes, which include time-dependent, irreversible processes such as that of conduction, chemical reactions and biological flow problems. Thus many authors in various forms have investigated the problem imposing different boundary conditions which suit the situations at hand and using various forms of f (x, t, u). The type of solution one gets depends on the form of f(x, t, u) and the boundary conditions. The solution u(x,t) can be defined for all positive t, in which case we call it a global solution or unbounded in finite time, in which case we say it blows up.

Escobedo and Herrero[1] and Fujita and Watanabe[2] considered

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
1(a)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

(b)


Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(c)

where, Ω is a bounded open subset of RN and established the existence and uniqueness of positive solution. Assuming that the initial value u0 is non-negative, they found the solution as:

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(2)

for all Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

Rossi[3] obtained the blow up rate for positive solution of:

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
3(a)

with boundary conditions

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(b)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(c)

In this case one has a non-linear term at the boundary and a reaction term in the equation. If λ>0, these 2 terms compete and blow up phenomenon occurs if p<2q-1 or p = 2q-1.

They established that if p<2q-1 or p = 2q-1 the blow up rate is given by:

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(4)

When λ>0, Chipot et al.[4] and Lopez et al.[5] proved the existence and regularity of solution for initial data that satisfies compatibility condition. They found out that the solution of eqn. (3) only exists for a finite period of time. Cazenave et al.[6] introduced the concave and convex term in the equation in the form:

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
5(a)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(b)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(c)

including a Dirichlet boundary condition. The non-linearity on the RHS is the sum of the concave and convex term with the non-linearity being singular at 0 (it is not Lipschitz because q<1). They showed that there exists a global solution if and only if there exists a weak solution of the stationary equation. Then Haraux[7] considered a linear parabolic equation with Lipschitz continuous boundary condition

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
6(a)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(b)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(c)

and showed that there is a unique global solution. Using this he established that for

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
7(a)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(b)

then

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(8)

In a recent work, Messaoudi[8] proved the local existence for the problem

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
9(a)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(b)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(c)

and showed that the solution blows up in finite time.

In the realm of the classical theory of differential equations, it can be shown imposing a Lipschitz condition on the non-linearity term f that the equation has a unique solution if u0(x) ∈ C (Ω). On the other hand if u0(x) is a distribution with compact support, the system has no solution in the classical sense[9]. Thus Ifidon and Oghre[10] formulated a generalized function G(Q), using classical estimates and induction hypothesis over the order of the differential operators which defines the element of G(Q) to prove the existence, uniqueness as well as consistence results for the solution to the problem.

This study considers:

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

and use an inverse process of Fourier transform to determine the solution u(x, t).

METHOD OF SOLUTION

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
10(a)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(b)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(c)

Let u(x, t) be the solution and assume that both u(x, t) and f(x, t) are piecewise smooth and integrable over (0, ∞), then by Fourier cosine transform

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(11)

Differentiating (11) with respect to t and substituting in (10)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(12)

where, Fc is the Fourier cosine transform of f(x, t).

Evaluating the integral on the right using integration by parts

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(13)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(14)

This is a linear differential equation whose integrating factor is given as

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
 
Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(15)

Taking the inverse Fourier cosine transform

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(16)

Interchanging the order of integration, the first integral becomes

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(17)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(18)

Using the fact that being odd then

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

Then equation (18) becomes

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(19)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(20)

Also the second integral in (16) becomes

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation
(21)

Hence by (19) and (21)

Image for - Fourier Transform Solution of the Semi-linear Parabolic Equation

CONCLUSION

We have considered the one dimensional semi-linear heat equation and we have considered a piecewise continuous integrable function f(x, t). Using a Fourier cosine transform and a backward inverse operation we have determined the solution to the problem.

REFERENCES
1:  Escobedo, M. and M.A. Herrero, 1993. A semi-linear parabolic system in a bounded domain. Ann. Matematica Pura Appl., 165: 315-336.
CrossRef  |  Direct Link  |  

2:  Fujita, H. and S. Watanabe, 1968. On the uniqueness of solutions of initial value problems for some quasi-linear parabolic equations. Commun. Pure Appl. Math., 21: 631-652.

3:  Rossi, J.D., 1998. The blow up rate for a semi-linear parabolic equation with a non-linear boundary condition. ACTA Math. Univ Comenianae, 67: 343-350.
Direct Link  |  

4:  Chipot, M., M. Fila and P. Quittner, 1991. Stationary solutions, blow up and convergence of stationary solutions for semi-linear parabolic equations with non-linear boundary conditions. Acta Math. Univ. Comenianae, 60: 35-103.
Direct Link  |  

5:  Lopez, G.J., V. Marquez and N. Wolanski, 1993. Dynamic behaviour of positive solutions to reaction-diffusion problems with nonlinear absoption through boundary. Rev. Union Math. Argentina, 38: 196-209.
Direct Link  |  

6:  Cazenave, T., F. Dickstein and M. Escobedo, 1999. A semi-linear heat equation with concave convex nonlinearity. Rendiconti Matematica, 19: 211-242.

7:  Haraux, A., 1997. A remark on parabolic equations. Portugaliae Math., 54: 311-316.

8:  Messaoudi, S., 2003. Local existence and blow up in a semi-linear heat equation with bessel operator. Nonlinear Stud., 10: 59-66.

9:  Brezis, H. and A. Friedman, 1983. Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pure Applied, 62: 73-97.

10:  Ifidon, E. and E. Oghre, 1999. Generalised solutions to non-linear parabolic equations. J. Nig. Math. Phys. Assoc., 3: 222-233.

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