Preface

This section discusses singular boundary value problems.

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Glossary

Singular Boundary Value Problems

Among the important applied problems in analytical methods is that of solving singular boundary value problems for differential equations. They arise naturally and repeatedly in physical models, often because of the coordinate system involved or because of an impulsive source or sink term.

We consider a class of singular boundary value problems arising in physiology:

\[ y'' + \frac{p}{x}\, y' = f(x,y) , \qquad y(0) =0 \mbox{ or } y' (0) =0, \quad \alpha\,y(\ell ) + \beta\,y' (\ell ) = 0 . \]
It is assumed that f(x, y) is continuously differentiable function of two variables. The singular boundary value problem arises in a number of applications, particularly for the cases when p = 0, 1, 2, and for certain linear and nonlinear functions f(x, y). Of special interest is the case when p = 2 and \( \displaystyle f(x,y) = \frac{a}{b+y} , \ a > 0, \ b > 0 , \) which arises in the modeling of steady state oxygen diffusion in a spherical cell with Michaelis--Menten uptake kinetics.

Another case of physical significance is when p = 2 and \( \displaystyle f(x,y) = \alpha\, e^{-ky} , \ k > 0, \ \alpha > 0 , \) which occurs in the formulation of the distribution of heat sources in the human head.

 

Example: Consider the singular boundary value problem

\[ y'' + \frac{1}{x}\, y' + e^{y(x)} = 0 , \qquad y(0) =0, \quad y(1) = 0 . \]
Its true solution is
\[ y(x) = 2\,\ln \frac{B+1}{B\,x^2 +1} , \qquad \box{where} \quad B = 3- 2\sqrt{2} . \]
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Example: Consider the singular boundary value problem

\[ y'' + \frac{2}{x}\, y' + \alpha\, e^{-k\,y(x)} = 0 , \qquad y(0) =0, \quad y(1) + 10\,y' (1) = 0 . \]
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Example: Consider the singular boundary value problem

\[ y'' + \frac{2}{x}\, y' = \frac{a\,y}{b+y} \qquad y' (0) =0, \quad 5\,y(1) + y' (1) = 6 . \]
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Example:

A mass balance on a differential volume element of porous medium for a spherical catalyst pellet gives a parabolic partial differential equation

\[ \frac{\partial c}{\partial t} = \nabla \cdot D_e \nabla c' + \left( - R_A \right) , \]
where t is time, c' is the chemical reactant concentration, (-RA) is the rate of reaction per unit volume, De is the effective diffusion coefficient for reactant, where the microscopic porous structure and other detail factors are incorporated together into De.

If we assume that the coupled process of diffusion and reaction taking place in the pores is at steady state and the variation in temperature in the spherical geometric pellet is negligible, hence we write

\[ D_e \left( \frac{\partial^2 c'}{\partial r^2} + \frac{2}{r}\,\frac{\partial c'}{\partial r} \right) = \left( -R_A \right) \]
with boundary conditions
\begin{align*} \frac{\partial c''}{\partial r} &= 0 \quad \mbox{at}\quad r=0 \quad \mbox{(center of catalyst))} \\ c' &= c_s \quad \mbox{at}\quad r=r_0 \quad \mbox{(surface of catalyst)} . \end{align*}

Now we consider the reaction A → B, with a rate which is the nth power of concentration of A, denoted as (-RA) = kC, where the reaction constant k is a function of temperature.

It is convenient to introduce dimensionless variables:

\[ R = \frac{r}{r_0} , \quad C=\frac{c''}{c_s} . \]
Then we have the Emden--Fowler equation
\[ \frac{\partial^2 C}{\partial R^2} + \frac{2}{R}\,\frac{\partial C}{\partial R} = \phi^2 C^n , \]
subject to the boundary conditions:
\[ \left. \frac{\partial C}{\partial R} \right\vert_{R=0} = 0 \qquad \mbox{and} \qquad C(1) = 1 , \]
where ϕ is the Thiele modulus \( \phi^2 = r_0^2 k\,c_s^{n-1} /D_e . \)

To simplify the solution procedure, we introduce a new variable \( Y= C\,R , \) and rewrite the goverming equation as

\[ \frac{{\text d}^2 Y}{{\text d} R^2} = \phi^2 Y^n R^{1-n} . \]
The corresponding boundary conditions become
\[ Y(0) = 0, \qquad Y(1) = 1 . \]
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Example: Consider the Dirichlet boundary value problem

\[ \frac{{\text d}^2 w}{{\text d} r^2} + \frac{1}{r}\,\frac{{\text d} w}{{\text d} r} - \frac{w}{r^2} =0 , \qquad \mbox{subject} \quad w(3) = 6, \quad w'(8) = 2. \]
Since the formula for the given boundary value problem is known to be
\[ w (r) = u(r) + \frac{2 - u(8)}{v(8)} \, v(r) , \]
we need to find u(r), the unique solution to the initial value problem
\[ \frac{{\text d}^2 u}{{\text d} r^2} + \frac{1}{r}\,\frac{{\text d} u}{{\text d} r} - \frac{u}{r^2} =0 , \qquad \mbox{subject} \quad u(3) = 6, \quad u'(3) = 0, \]
and v(r), which is the solution to IVP:
\[ \frac{{\text d}^2 v}{{\text d} r^2} + \frac{1}{r}\,\frac{{\text d} v}{{\text d} r} - \frac{v}{r^2} =0 , \qquad \mbox{subject} \quad v(3) = 0, \quad v'(3) = 1. \]
The given differential equation is the Euler one, so its general solution is
\[ u(r) = A\,r + \frac{B}{r} \qquad \mbox{and} \qquad v(r) = a\, r + \frac{b}{r}. \]
To find the values of constants, we substitute their forms into the initial conditions:
\[ \begin{split} u(3) = A\, 3 + \frac{B}{3} =1, \quad u' (3) = A - \frac{B}{9} =0 , \\ v(3) = a\,3 + \frac{b}{3} =0, \quad a- \frac{b}{9} =1 . \end{split} \]
Therefore,
\[ u(r) = r + \frac{9}{r} \qquad \mbox{and} \qquad v(r) = \frac{1}{2}\, r - \frac{9}{2\,r}. \]
Since u(8) = 73/8 and v(8) = 55/16, we find the required solution
\[ w (r) = r + \frac{9}{r} + \frac{2 - 73/8}{55/16} \left( \frac{1}{2}\, r - \frac{9}{2\,r} \right) = \frac{2}{55\,r} \left( 504 - r^2 \right) , \]
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  1. Azimi, M., Azimi, A., Investigation on Reaction Diffusion Process Inside a Porous Bio-Catalyst Using DTM, Journal of Bioequivalence & Bioavailability, 2015, Vol. 7, No. 3; doi: 10.4172/jbb.1000225
  2. Duan, J.-S. and Rach, R., The degenerate form of the Adomian polynomials in the power series method for nonlinear ordinary differential equations, Journal of Mathematics and System Science, Volume 5, Pages 411--428, doi: 10.17265/2159-5291/2015.10.003
  3. Magyari, E., Exact analytical solutions of diffusion reaction in spherical porous catalyst, Chemical Engineering Journal, 2010, Volume 158, Issue 2, 1 April 2010, Pages 266-270; https://doi.org/10.1016/j.cej.2010.01.034
  4. Mittal, R.C., and Nigam, R., “Solution of a class of singular boundaryvalue problems”, Numerical Algorithms, 2008, vol. 47, pp. 169–179, 2008.
  5. Versypt, A.N., Arendt, P.D., Braatz, R.D., Derivation of an Analytical Solution to a Reaction-Diffusion Model for Autocatalytic Degradation and Erosion in Polymer Microspheres, PLOS,
  6. Yan, B., Kiu, Y., Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line, Journal Applied Mathematics and Computation, 2004, Volume 147 Issue 3, pp. 629--644; doi: 10.1016/S0096-3003(02)00801-9

 

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