**2. Signal-dependent sensitivity model**

Let us start with a model that allows nonnegative bounded solutions that may be of interest from a biological point of view. Now consider the "logistic" model, one of versions of signal-dependent sensitivity model [8] with the chemosensitivity functions *<sup>ϕ</sup>*ð Þ¼ *<sup>v</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>b</sup>* ln ð Þ *<sup>v</sup>* <sup>þ</sup> *<sup>b</sup>* , where *<sup>b</sup>* <sup>¼</sup> *const*, and *f u*ð Þ¼ , *<sup>v</sup> <sup>σ</sup>*~*<sup>u</sup>* � <sup>~</sup>*βv*. In the review [5] one can see a mathematical analysis of this model. When *<sup>b</sup>* <sup>¼</sup> 0 and *<sup>β</sup>*<sup>~</sup> <sup>¼</sup> 0, the existence of traveling waves was established in [16, 17]. The replacements of *<sup>t</sup>* ! *<sup>δ</sup>*1*<sup>t</sup>* and *<sup>u</sup>* ! *<sup>σ</sup> <sup>σ</sup>*<sup>~</sup> *δ*1 *<sup>u</sup>* give *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> 1, *<sup>α</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>2</sup> *δ*1 , *<sup>β</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *δ*1 , and *<sup>σ</sup>* ¼ �1. We also set *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1ð Þ <sup>1</sup>þ*<sup>b</sup> <sup>δ</sup>*<sup>1</sup> , 1 þ *b*>0, as well as *ϕ*ð Þ¼ *v* ln ∣*v* þ *b*∣. It should be noted that a sign of *σ* may effect on the mathematical properties of the system. So, *σ* ¼ 1 corresponds to an increase of a chemical substance, proportional to cell density, whereas *σ* ¼ �1 corresponds to its decrease. And as we shall see later, various solutions correspond to these two cases.

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

After the above replacements, the model reads:

$$\begin{cases} u\_t - u\_{\text{xx}} + \eta \left( u \frac{v\_x}{\nu + b} \right)\_x = 0 \\ v\_t - a v\_{\text{xx}} - \sigma u + \beta v = 0. \end{cases} \tag{1}$$

If we introduce the function *υ* ¼ *v* þ *b*, in terms of traveling wave variable *y* ¼ *x* � *ct*, where *c* ¼ *const*, this system has the form:

$$\begin{cases} \left. u\_{\mathcal{V}} + c\mu - \eta u (\ln \left( \nu \right))\_{\mathcal{y}} + \lambda = 0 \\ \left. a v\_{\mathcal{Y}\mathcal{Y}} + c v\_{\mathcal{Y}} - \beta \nu + \beta b + \sigma u = 0, \end{cases} \right. \tag{2}$$

where *u* ¼ *u y* ð Þ, *υ* ¼ *υ*ð Þ*y* , and *λ* is an integration constant. In this chapter we will consider the case of *λ* ¼ 0. Then Eq. (2) gives:

$$
\mu = \mathbb{C}\_{\mu} e^{-\epsilon \mathcal{y}} v^{\eta}, \tag{3}
$$

*Cu* is a constant and we will examine the following equation for *υ*:

$$
\rho a v\_{\mathcal{Y}} + \varepsilon v\_{\mathcal{Y}} - \beta v + \beta b + \sigma \mathcal{C}\_u e^{-c\mathcal{Y}} v^\eta = 0. \tag{4}
$$

Since *η* is a positive constant, we consider two cases: *η* ¼ 1 [Eq. (4) is a linear nonhomogeneous equation] and *η* 6¼ 1.

A. *η* ¼ 1

introduction into the mathematics of the Patlak-Keller-Segel model and summarizes different mathematical results; the detailed reviews also can be found in the textbooks of Suzuki [6] and Perthame [7]. In the review of Hillen and Painter [8], a number of variations of the original Patlak-Keller-Segel model are explored in detail. The authors study their formulation from a biological perspective, summarize key results on their analytical properties, and classify their solution forms [8]. It should be noted that interest in the Patlak-Keller-Segel model does not weaken and new works appear devoted to the study of various properties of equations and their

*Mathematical Theorems - Boundary Value Problems and Approximations*

In this chapter we investigate a number of different models describing chemotaxis. The aim of this paper is to obtain exact analytical solutions of these models. For one-dimensional parabolic-parabolic systems under consideration, we present these solutions in explicit form in terms of traveling wave variables. Of course, not all of the solutions obtained can have appropriate biological interpretation since the biological functions must be nonnegative in all domains of definition. However some of these solutions are positive and bounded, and their analysis requires further investigation. Despite the large number of works devoted to the systems under consideration and their properties, as well as the properties of their solutions, it

The Patlak-Keller-Segel model describes the space–time evolution of a cell

*ut* � ∇ð*δ*1∇*u* � *η*1*u*∇*ϕ*ð Þ*v* Þ ¼ 0

where *δ*<sup>1</sup> >0 and *δ*<sup>2</sup> ≥0 are cell and chemical substance diffusion coefficients, respectively, and *η*<sup>1</sup> is a chemotaxis coefficient; when *η*<sup>1</sup> >0, this is an attractive chemotaxis ("positive taxis"), and when *η*<sup>1</sup> < 0, this is a repulsive ("negative") one [13, 14]. *ϕ*ð Þ*v* is the chemosensitivity function, and *f u*ð Þ , *v* characterizes the chemical growth and degradation. These functions are taken in different forms that correspond to some variations of the original Patlak-Keller-Segel model. We follow the reviews of Hillen and Painter [8] and of Wang [15] and consider the models

This paper is concerned with one-dimensional simplified models when the coefficients *δ*1, *δ*2, and *η*<sup>1</sup> are positive constants, *x*∈ ℜ, *t* ≥0, *u* ¼ *u x*ð Þ , *t* , and *v* ¼ *v x*ð Þ , *t* .

Let us start with a model that allows nonnegative bounded solutions that may be of interest from a biological point of view. Now consider the "logistic" model, one of versions of signal-dependent sensitivity model [8] with the chemosensitivity functions *<sup>ϕ</sup>*ð Þ¼ *<sup>v</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>b</sup>* ln ð Þ *<sup>v</sup>* <sup>þ</sup> *<sup>b</sup>* , where *<sup>b</sup>* <sup>¼</sup> *const*, and *f u*ð Þ¼ , *<sup>v</sup> <sup>σ</sup>*~*<sup>u</sup>* � <sup>~</sup>*βv*. In the review [5] one can see a mathematical analysis of this model. When *<sup>b</sup>* <sup>¼</sup> 0 and *<sup>β</sup>*<sup>~</sup> <sup>¼</sup> 0, the existence of traveling waves was established in [16, 17]. The replacements of

> *δ*1 , *<sup>β</sup>* <sup>¼</sup> *<sup>β</sup>*<sup>~</sup> *δ*1

1 þ *b*>0, as well as *ϕ*ð Þ¼ *v* ln ∣*v* þ *b*∣. It should be noted that a sign of *σ* may effect on the mathematical properties of the system. So, *σ* ¼ 1 corresponds to an increase of a chemical substance, proportional to cell density, whereas *σ* ¼ �1 corresponds to its decrease. And as we shall see later, various solutions correspond to these

*v* � *f u*ð Þ¼ , *v* 0,

!Þ. The general form

, and *<sup>σ</sup>* ¼ �1. We also set *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1ð Þ <sup>1</sup>þ*<sup>b</sup>*

*<sup>δ</sup>*<sup>1</sup> ,

!Þ and a concentration of a chemical substance *<sup>v</sup>*ð*t*, *<sup>r</sup>*

seems to us that the solutions obtained in this paper are new.

*vt* � *<sup>δ</sup>*2∇<sup>2</sup>

**2. Signal-dependent sensitivity model**

*δ*1

*<sup>u</sup>* give *<sup>δ</sup>*<sup>1</sup> <sup>¼</sup> 1, *<sup>α</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>2</sup>

solutions [9–12] and the links below.

density *u*ð*t*, *r*

of this model is:

presented therein.

*<sup>t</sup>* ! *<sup>δ</sup>*1*<sup>t</sup>* and *<sup>u</sup>* ! *<sup>σ</sup> <sup>σ</sup>*<sup>~</sup>

two cases.

**2**

Let us begin with *η* ¼ 1. We introduce the new variable *z* and the new function *w*:

$$\begin{aligned} z &= \left(\frac{4\sigma \mathbf{C}\_u}{ac^2}\right)^{\frac{1}{2}} e^{-\frac{\sigma}{2}}\\ w &= \left(\frac{4\sigma \mathbf{C}\_u}{ac^2}\right)^{\frac{a-2}{4a}} v e^{\frac{\sigma}{2a}} \end{aligned} \tag{5}$$

and Eq. (4) becomes:

$$z^2 w\_{xx} + z w\_x + w \left(z^2 - \nu^2\right) = \Lambda z^{-\frac{1}{a}},\tag{6}$$

where *<sup>ν</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *<sup>α</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>*αβ c*2 � � and <sup>Λ</sup> ¼ � <sup>4</sup>*β<sup>b</sup> αc*<sup>2</sup> 4*σCu αc*<sup>2</sup> � �<sup>1</sup> 4 . Eq. (6) is the Lommel differential equation [18, 19] with *<sup>μ</sup>* ¼ �<sup>1</sup> � <sup>1</sup> *<sup>α</sup>*, and we consider *σCu* > 0. Since this is a linear inhomogeneous second-order differential equation, one can integrate it by the method of variation of parameters. We assume a solution in the form:

$$w(z) = \mathcal{C}\_l(z) f\_\nu(z) + \mathcal{C}\_Y(z) Y\_\nu(z),$$

where *Jν*ð Þ*z* and *Yν*ð Þ*z* are Bessel functions and *CJ*ð Þ*z* and *CY*ð Þ*z* are the functions of *z* that satisfy the equations:

$$\begin{aligned} \left(J\_{\nu}(\boldsymbol{z})\left(\mathbf{C}\_{l}(\boldsymbol{z})\right)\_{\boldsymbol{x}} + Y\_{\nu}(\boldsymbol{z})\left(\mathbf{C}\_{Y}(\boldsymbol{z})\right)\_{\boldsymbol{x}} = \mathbf{0} \\ \left(J\_{\nu}(\boldsymbol{z})\right)\_{\boldsymbol{x}}\left(\mathbf{C}\_{l}(\boldsymbol{z})\right)\_{\boldsymbol{x}} + \left(Y\_{\nu}(\boldsymbol{z})\right)\_{\boldsymbol{x}}\left(\mathbf{C}\_{Y}(\boldsymbol{z})\right)\_{\boldsymbol{x}} = \boldsymbol{\Lambda}\boldsymbol{z}^{-\frac{1}{\sigma}}. \end{aligned}$$

Considering that Wronskian *W J<sup>ν</sup>* ð Þ , *<sup>Y</sup><sup>ν</sup>* ð Þ¼ *<sup>z</sup>* <sup>2</sup> *πz* , we obtain: *Mathematical Theorems - Boundary Value Problems and Approximations*

$$\mathcal{C}\_{I}(z) = c\_{I} - \frac{\Lambda \pi}{2} \left[ z^{-1 - \frac{1}{a}} Y\_{\nu}(z) dz \right]$$

$$\mathcal{C}\_{Y}(z) = c\_{Y} + \frac{\Lambda \pi}{2} \left[ z^{-1 - \frac{1}{a}} J\_{\nu}(z) dz, \right]$$

where *cJ* and *cY* are constants. If both of the numbers � <sup>1</sup> *<sup>α</sup>* � *ν* are positive, the lower limits in the integrals may be taken to be zero. Then a particular integral of Lommel equation "proceeding in ascending powers of *z*" is *s<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* [19]; if one considers a solution of Lommel equation "in the form of descending series," one obtains the function *S<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* [19] [see Eq. (8)]. Thus, quoting Watson [19] "...and so, of Lommel's two functions *s<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* and *S<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* , it is frequently more convenient to use the latter." Then the general solution of Eq. (6) has the form:

$$w(z) = C\_l f\_\nu(z) + C\_Y Y\_\nu(z) + \Lambda S\_{\mu,\nu}(z),\tag{7}$$

where *CJ* and *CY* are constants,

$$S\_{\mu,\nu}(z) = s\_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu-\nu+1}{2}\right) \Gamma\left(\frac{\mu+\nu+1}{2}\right)$$

$$\left[\sin\left(\frac{\pi}{2}(\mu-\nu)\right)I\_{\nu}(z) - \cos\left(\frac{\pi}{2}(\mu-\nu)\right)Y\_{\nu}(z)\right],\tag{8}$$

$$s\_{\mu,\nu}(z) = \frac{z^{\mu+1}}{\left[\left(\mu+1\right)^2 - \nu^2\right]} \,\_1F\_2\left(1; \frac{\mu-\nu+3}{2}, \frac{\mu+\nu+3}{2}; -\frac{z^2}{4}\right)$$

are Lommel functions, and <sup>1</sup>*F*<sup>2</sup> is the generalized hypergeometric function [18, 19]. Further, substituting the initial variable *y* and the function *v* [see Eq. (5)] into Eq. (7), we obtain a formal solution.

1.*b* ¼ 0

We first consider the case *b* ¼ 0. Then *υ* ¼ *v*≥ 0 and *Cu* >0. Eq. (6) becomes homogeneous, and for *σ* ¼ 1, its general solution is:

$$
\Delta w(\mathbf{z}) = \mathbf{C}\_{\text{J}} I\_{\nu}(\mathbf{z}) + \mathbf{C}\_{\text{Y}} Y\_{\nu}(\mathbf{z}). \tag{9}
$$

the plots for *c* ¼ 5 are thicker than for *c* ¼ 1. Thus, the solution obtained may be considered as a biologically appropriated one, and this requires further investigation.

(a) (b)

0; 1 3; 1 2;

Out[44]=

0; 1 3; 1 2; Out[27]=

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

(a) (b)

Let us return to Eq. (6) with Λ 6¼ 0. The analysis of solution asymptotic forms

Thus, one can see that for *b*> 0, *σ* ¼ 1, and *Cu* > 0, *u y* ð Þ≥0 is satisfied but *v y* ð Þ<0.

Using the analysis of Eq. (11), one can see that the condition *b*<0 along with

ffiffiffiffiffiffiffiffiffiffiffi 4*σCu αc*<sup>2</sup>

r

!

*e* �*cy* 2

5 0 5 10 15 20 25 30

1.5 1.0 0.5 0.5 1.0 1.5

2 4 6

u y

v y

0.5 1.0 1.5 2.0

> *e* �*cy* 2

(11)

0; 1 3; 1 2;

0; 1 3; 1 2;

<sup>4</sup>*<sup>α</sup>* ≤*β* <0.

ffiffiffiffiffiffiffiffiffiffiffi 4*σCu αc*<sup>2</sup>

!

r

�*cy* <sup>1</sup><sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*<sup>α</sup> S<sup>μ</sup>*,*<sup>ν</sup>*

*<sup>α</sup>*. The latter condition leads to the requirement � *<sup>c</sup>*<sup>2</sup>

*<sup>σ</sup>* as *cy* ! �∞ and *v y* ð Þ! 0 and *u y* ð Þ! 0 as *cy* ! ∞.

4*σCu αc*<sup>2</sup> � �<sup>1</sup> 2*α e* �*cy* <sup>2</sup>*<sup>α</sup> Sμ*,*<sup>ν</sup>*

at �∞ [18, 19] gives the following expressions for *v y* ð Þ and *u y* ð Þ:

*αc*<sup>2</sup>

4*σCu αc*<sup>2</sup> � �<sup>1</sup> 2*α e*

These functions are presented in **Figures 3** and **4**. It should be noted that

*σ* ¼ �1 and *Cu* <0 (*σCu* >0) leads to the fact that the function *u y* ð Þ has not changed, but *v y* ð Þ becomes positive on all domains of definition. This function is

4*βb αc*<sup>2</sup>

*v y* ð Þþ *<sup>b</sup>* ¼ � <sup>4</sup>*β<sup>b</sup>*

*<sup>α</sup>* or *β* 6¼ 0 because of the pole in Γ function.

*u y* ð Þ¼�*Cu*

5 0 5 10 15 20 25 30

u y

5 5 10 15

*(a) u y* ð Þ*; c* ¼ *1; Cu* ¼ *18. (b) u y* ð Þ*; c* ¼ *5; Cu* ¼ *18.*

0.02 0.04 0.06 0.08 0.10 0.12 0.14

*(a) v y* ð Þ*; c* ¼ *1; c* ¼ *5; Cu* ¼ *18. (b) v y* ð Þ*; c* ¼ *1; c* ¼ *5; Cu* ¼ *2.*

v y

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

with *σCu* >0 and *ν*< <sup>1</sup>

The *v y* ð Þ!�*b*, *u y* ð Þ!� *<sup>β</sup><sup>b</sup>*

2.*b*> 0

Out[23]=

**Figure 1.**

Out[36]=

**Figure 2.**

0.2 0.4 0.6 0.8 1.0 1.2

*<sup>ν</sup>* 6¼ <sup>1</sup>

**5**

3.*b*< 0

presented in **Figure 5**.

However one can check that the function *u* ¼ *u y* ð Þ diverges as *cy* ! �∞ for all *ν*.

Consider now *σ* ¼ �1. For *v y* ð Þ to be real, let *α* ¼ 2. Then Eq. (6) becomes the modified Bessel equation; the analysis of solution behavior at �∞ leads to suitable solutions for *v y* ð Þ and *u y* ð Þ:

$$\begin{aligned} v(\boldsymbol{y}) &= e^{-\frac{\boldsymbol{\sigma}}{4}} K\_{\nu} \left( \sqrt{\frac{2\mathbf{C}\_{u}}{c^{2}}} e^{-\frac{\boldsymbol{\sigma}}{2}} \right) \\ u(\boldsymbol{y}) &= \mathbf{C}\_{u} e^{-\frac{\boldsymbol{\sigma}\mathbf{y}}{4}} K\_{\nu} \left( \sqrt{\frac{2\mathbf{C}\_{u}}{c^{2}}} e^{-\frac{\boldsymbol{\sigma}}{2}} \right) \end{aligned} \tag{10}$$

with restrictions *ν*≤ <sup>1</sup> <sup>2</sup> and *β* ≤0. So one can see that *v y* ð Þ! 0 as *cy* ! �∞ for all *ν*≤ <sup>1</sup> 2 ; *v y* ð Þ! 0 for *<sup>ν</sup>*<sup>&</sup>lt; <sup>1</sup> <sup>2</sup> and *v y* ð Þ! ffiffiffiffiffiffi *π*2*c*<sup>2</sup> 8*Cu* 4 q for *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> as *cy* ! ∞ and *u y* ð Þ! 0 as *y* ! �<sup>∞</sup> for all *<sup>ν</sup>*<sup>≤</sup> <sup>1</sup> 2 . The curves of these functions are presented in **Figures 1** and **2**, and *Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

**Figure 1.** *(a) v y* ð Þ*; c* ¼ *1; c* ¼ *5; Cu* ¼ *18. (b) v y* ð Þ*; c* ¼ *1; c* ¼ *5; Cu* ¼ *2.*

**Figure 2.** *(a) u y* ð Þ*; c* ¼ *1; Cu* ¼ *18. (b) u y* ð Þ*; c* ¼ *5; Cu* ¼ *18.*

the plots for *c* ¼ 5 are thicker than for *c* ¼ 1. Thus, the solution obtained may be considered as a biologically appropriated one, and this requires further investigation.

2.*b*> 0

*CJ*ð Þ¼ *<sup>z</sup> cJ* � <sup>Λ</sup>*<sup>π</sup>*

*Mathematical Theorems - Boundary Value Problems and Approximations*

*CY*ð Þ¼ *z cY* þ

where *cJ* and *cY* are constants. If both of the numbers � <sup>1</sup>

the latter." Then the general solution of Eq. (6) has the form:

where *CJ* and *CY* are constants,

*sμ*,*<sup>ν</sup>*ð Þ¼ *z*

1.*b* ¼ 0

solutions for *v y* ð Þ and *u y* ð Þ:

with restrictions *ν*≤ <sup>1</sup>

; *v y* ð Þ! 0 for *<sup>ν</sup>*<sup>&</sup>lt; <sup>1</sup>

2

*ν*≤ <sup>1</sup> 2

**4**

�<sup>∞</sup> for all *<sup>ν</sup>*<sup>≤</sup> <sup>1</sup>

into Eq. (7), we obtain a formal solution.

*<sup>S</sup>μ*, *<sup>ν</sup>*ð Þ¼ *<sup>z</sup> <sup>s</sup>μ*,*<sup>ν</sup>*ð Þþ *<sup>z</sup>* <sup>2</sup>*<sup>μ</sup>*�<sup>1</sup>

sin *<sup>π</sup>* 2 ð Þ *μ* � *ν* � �

homogeneous, and for *σ* ¼ 1, its general solution is:

*v y* ð Þ¼ *e*

*u y* ð Þ¼ *Cue*

<sup>2</sup> and *v y* ð Þ!

�*cy* <sup>4</sup> *K<sup>ν</sup>*

> �5*cy* <sup>4</sup> *K<sup>ν</sup>*

ffiffiffiffiffiffi *π*2*c*<sup>2</sup> 8*Cu* 4 q

*zμ*þ<sup>1</sup> ð Þ *<sup>μ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> � *<sup>ν</sup>*<sup>2</sup> 2 ð *z*�1�<sup>1</sup>

Λ*π* 2 ð *z*�1�<sup>1</sup>

lower limits in the integrals may be taken to be zero. Then a particular integral of Lommel equation "proceeding in ascending powers of *z*" is *s<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* [19]; if one considers a solution of Lommel equation "in the form of descending series," one obtains the function *S<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* [19] [see Eq. (8)]. Thus, quoting Watson [19] "...and so, of Lommel's two functions *s<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* and *S<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* , it is frequently more convenient to use

> <sup>Γ</sup> *<sup>μ</sup>* � *<sup>ν</sup>* <sup>þ</sup> <sup>1</sup> 2 � �

h i <sup>1</sup>*F*<sup>2</sup> 1; *<sup>μ</sup>* � *<sup>ν</sup>* <sup>þ</sup> <sup>3</sup>

are Lommel functions, and <sup>1</sup>*F*<sup>2</sup> is the generalized hypergeometric function [18, 19]. Further, substituting the initial variable *y* and the function *v* [see Eq. (5)]

We first consider the case *b* ¼ 0. Then *υ* ¼ *v*≥ 0 and *Cu* >0. Eq. (6) becomes

However one can check that the function *u* ¼ *u y* ð Þ diverges as *cy* ! �∞ for all *ν*. Consider now *σ* ¼ �1. For *v y* ð Þ to be real, let *α* ¼ 2. Then Eq. (6) becomes the modified Bessel equation; the analysis of solution behavior at �∞ leads to suitable

> ffiffiffiffiffiffiffiffi 2*Cu c*2

r

for *<sup>ν</sup>* <sup>¼</sup> <sup>1</sup>

. The curves of these functions are presented in **Figures 1** and **2**, and

!

*e* �*cy* 2

> *e* �*cy* 2

<sup>2</sup> and *β* ≤0. So one can see that *v y* ð Þ! 0 as *cy* ! �∞ for all

! (10)

<sup>2</sup> as *cy* ! ∞ and *u y* ð Þ! 0 as *y* !

ffiffiffiffiffiffiffiffi 2*Cu c*2

r

*<sup>J</sup>ν*ð Þ� *<sup>z</sup>* cos *<sup>π</sup>*

h i

*<sup>α</sup>Yν*ð Þ*z dz*

*<sup>α</sup> Jν*ð Þ*z dz*,

*w z*ð Þ¼ *CJJν*ð Þþ *z CYYν*ð Þþ *z* Λ*S<sup>μ</sup>*,*<sup>ν</sup>*ð Þ*z* , (7)

<sup>Γ</sup> *<sup>μ</sup>* <sup>þ</sup> *<sup>ν</sup>* <sup>þ</sup> <sup>1</sup> 2 � �

*Yν*ð Þ*z*

*μ* þ *ν* þ 3

� �

*w z*ð Þ¼ *CJJν*ð Þþ *z CYYν*ð Þ*z :* (9)

,

4

(8)

<sup>2</sup> ; � *<sup>z</sup>*<sup>2</sup>

2 ð Þ *μ* � *ν* � �

<sup>2</sup> ,

*<sup>α</sup>* � *ν* are positive, the

Let us return to Eq. (6) with Λ 6¼ 0. The analysis of solution asymptotic forms at �∞ [18, 19] gives the following expressions for *v y* ð Þ and *u y* ð Þ:

$$\begin{split} v(\mathbf{y}) + b &= -\frac{4\beta b}{ac^2} \left( \frac{4\sigma \mathbf{C}\_u}{ac^2} \right)^{\frac{1}{2a}} e^{-\frac{\gamma}{2a}} S\_{\mu,\nu} \left( \sqrt{\frac{4\sigma \mathbf{C}\_u}{ac^2}} e^{-\frac{\gamma}{2}} \right) \\ u(\mathbf{y}) &= -\mathbf{C}\_u \frac{4\beta b}{ac^2} \left( \frac{4\sigma \mathbf{C}\_u}{ac^2} \right)^{\frac{1}{2a}} e^{-\gamma \left( 1 + \frac{1}{2a} \right)} S\_{\mu,\nu} \left( \sqrt{\frac{4\sigma \mathbf{C}\_u}{ac^2}} e^{-\frac{\gamma}{2}} \right) \end{split} \tag{11}$$

with *σCu* >0 and *ν*< <sup>1</sup> *<sup>α</sup>*. The latter condition leads to the requirement � *<sup>c</sup>*<sup>2</sup> <sup>4</sup>*<sup>α</sup>* ≤*β* <0. The *v y* ð Þ!�*b*, *u y* ð Þ!� *<sup>β</sup><sup>b</sup> <sup>σ</sup>* as *cy* ! �∞ and *v y* ð Þ! 0 and *u y* ð Þ! 0 as *cy* ! ∞. Thus, one can see that for *b*> 0, *σ* ¼ 1, and *Cu* > 0, *u y* ð Þ≥0 is satisfied but *v y* ð Þ<0. These functions are presented in **Figures 3** and **4**. It should be noted that *<sup>ν</sup>* 6¼ <sup>1</sup> *<sup>α</sup>* or *β* 6¼ 0 because of the pole in Γ function.

3.*b*< 0

Using the analysis of Eq. (11), one can see that the condition *b*<0 along with *σ* ¼ �1 and *Cu* <0 (*σCu* >0) leads to the fact that the function *u y* ð Þ has not changed, but *v y* ð Þ becomes positive on all domains of definition. This function is presented in **Figure 5**.

and we consider the case *b* ¼ 0. Thus, *υ* ¼ *v*, and for:

we obtain the Abel equation of the second kind:

*wz* ½ð Þ <sup>1</sup> � *<sup>η</sup> <sup>w</sup>* � *<sup>α</sup>z*� þ ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>η</sup>* � <sup>1</sup> *<sup>z</sup>*�<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

*w* ¼ *z* 2�*α* <sup>2</sup> *t* þ

2*σαCu c*2 � �� *<sup>η</sup>*þ<sup>3</sup>

*c*ð Þ 2*α* þ *η* � 1

2 2ð Þ *α* þ *η* � 1

2 2ð Þ *α* þ *η* � 1

*<sup>C</sup><sup>ϑ</sup>* <sup>&</sup>gt;∣*C*~*ϑ*<sup>∣</sup> *<sup>π</sup>*

! <sup>2</sup>

! <sup>2</sup>

*y t*ðÞ¼� *α η*ð Þ <sup>þ</sup> <sup>3</sup>

*v t*ðÞ¼ � *<sup>C</sup>*<sup>~</sup> *<sup>ϑ</sup>*ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>3</sup>

*u t*ðÞ¼ *Cu* � *<sup>C</sup>*~*ϑ*ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>3</sup>

function as *t* ! �∞ [18], we can take:

2ð Þ *η*þ1 *t*2*F*<sup>1</sup> 1 2

leads to:

where we take

*<sup>ϑ</sup>*ðÞ¼ *<sup>t</sup> <sup>C</sup>*~*<sup>ϑ</sup>*

we obtain:

**7**

*<sup>z</sup>* <sup>¼</sup> *<sup>v</sup>* 1�*η α y*

Then we find the solutions of Eq. (15) in parametric form [21] with the parameter *t*. Now we consider the case 2*α* þ *η* 6¼ 1. A combination of substitutions

*<sup>z</sup>* ¼ � ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>3</sup> ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>1</sup> *<sup>t</sup>*

*<sup>w</sup>* <sup>¼</sup> *vy <sup>v</sup>*�*α*þ*η*�<sup>1</sup>

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

*α*

*<sup>w</sup>*<sup>2</sup> <sup>þ</sup> *<sup>α</sup><sup>z</sup>* � *αβ*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*σαCu c*2

> *α* 2

3 2

� �

� �

!<sup>2</sup>

2 2ð Þ *α* þ *η* � 1

2 2ð Þ *α* þ *η* þ 1 ð Þ *<sup>η</sup>* � <sup>1</sup> ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>3</sup> *<sup>z</sup>*

and Eq. (15) becomes an equation for the function *ϑ*ð Þ*t* . Solving it, for *σCu* >0,

, *<sup>η</sup>* <sup>þ</sup> <sup>3</sup> <sup>2</sup>ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>1</sup> ;

where *C*~*<sup>ϑ</sup>* and *C<sup>ϑ</sup>* are constants and <sup>2</sup>*F*<sup>1</sup> is the hypergeometric Gauss function.

Further we obtain the solutions of initial Eqs. (3)–(4) in parametric form:

ð Þ *η* þ 1 *t*

ð Þ *η* þ 1 *t*

where the constant *<sup>C</sup>*~*<sup>ϑ</sup>* is chosen so that 2ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>η</sup>* � <sup>1</sup> *<sup>C</sup>*<sup>~</sup> *<sup>ϑ</sup>* <sup>&</sup>lt;0, which is consistent

2*σαCu c*2 � �� <sup>1</sup>

with Eq. (17). Using the asymptotic representation of hypergeometric Gauss

2 þ

� �� <sup>1</sup>

2 þ

� �� <sup>1</sup>

2*σαCu c*2

> 2*σαCu c*2

> > *<sup>η</sup>*þ<sup>1</sup> Γ <sup>1</sup> *η*þ1 � �

> > > Γ *<sup>η</sup>*þ<sup>3</sup> 2ð Þ *η*þ1

ln ð Þ *ϑ*ð Þ*t*

1�*η*

1�*η*

2 ffiffiffiffiffiffiffiffiffiffi *<sup>η</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup>

� �

*<sup>c</sup>*<sup>2</sup> <sup>þ</sup> *σαCu*

� �

*ϑt*ð Þ*t ϑ*ð Þ*t*

<sup>þ</sup> *<sup>α</sup>* 1 � *η z*,

*ϑ*ð Þ*t* >0 and 2ð Þ *α* þ *η* � 1 *ϑt*ð Þ*t* <0, (17)

; � ð Þ *<sup>η</sup>* <sup>þ</sup> <sup>1</sup> *<sup>c</sup>*<sup>2</sup> 2*σαCu*

*η*þ1

*η*þ1

*t* 2

ð Þ *<sup>ϑ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup>�*<sup>α</sup>* <sup>2</sup>*α*þ*η*�<sup>1</sup>

ð Þ *<sup>ϑ</sup>*ð Þ*<sup>t</sup> αη*þ2*α*þ<sup>2</sup> 2*α*þ*η*�1

� � (20)

þ *Cϑ*, (18)

(19)

*<sup>c</sup>*<sup>2</sup> *<sup>z</sup>*�*<sup>α</sup>*

*α*

(14)

(16)

¼ 0*:* (15)

**Figure 4.**

*(a) u y* ð Þ*; c* ¼ *1; Cu* ¼ *9; σ* ¼ *1; b* ¼ *0:1. (b) u y* ð Þ*; c* ¼ *1; σ* ¼ *1; b* ¼ *0:1; α* ¼ *1; β* ¼ �*1=4; ν* ¼ *0.*

**Figure 5.** *v y* ð Þ*; c* ¼ *1; Cu* ¼ �*9; σ* ¼ �*1; b* ¼ �*0:1.*

B. *η* 6¼ 1

Let us return to Eq. (4) and rewrite it in terms of the variable *<sup>ξ</sup>* <sup>¼</sup> *<sup>e</sup>*�*cy α* :

$$
\xi^2 \nu\_{\xi\xi} - \frac{a\beta}{c^2} \nu + \frac{\sigma a \mathcal{C}\_u}{c^2} \,\, \xi^a \,\nu^\eta = -\frac{a\beta b}{c^2} \,. \tag{12}
$$

To integrate this equation, we use the Lie group method of infinitesimal transformations [20]. We find a group invariant of a second prolongation of oneparameter symmetry group vector of (12), and with its help, we transform Eq. (12) into an equation of the first order. It turns out that nontrivial symmetry group requires some conditions:

$$\begin{aligned} \frac{a\beta b}{c^2} &= \mathbf{0},\\ \beta &= \frac{(a-2)(a+\eta+\mathbf{1})c^2}{a(\eta+\mathbf{3})^2} \end{aligned} \tag{13}$$

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

and we consider the case *b* ¼ 0. Thus, *υ* ¼ *v*, and for:

$$\begin{aligned} z &= \frac{v^{\frac{1-q}{a}}}{y} \\ w &= v\_{\mathcal{Y}} v^{-\frac{a+q-1}{a}} \end{aligned} \tag{14}$$

we obtain the Abel equation of the second kind:

$$w\_x \left[ (\mathbf{1} - \eta)w - a\mathbf{z} \right] + (a + \eta - \mathbf{1})\mathbf{z}^{-1}w^2 + a\mathbf{z} \left( -\frac{a\beta}{c^2} + \frac{\sigma a \mathbf{C}\_u}{c^2} \mathbf{z}^{-a} \right) = \mathbf{0}.\tag{15}$$

Then we find the solutions of Eq. (15) in parametric form [21] with the parameter *t*. Now we consider the case 2*α* þ *η* 6¼ 1. A combination of substitutions leads to:

$$\begin{aligned} z &= \left( -\frac{(\eta+3)\left[ (\eta+1)t^2 + \frac{2\sigma a C\_u}{c^2} \right]}{2(2a+\eta-1)} \frac{\theta\_t(t)}{\theta(t)} \right)^{\frac{2}{a}} \\ w &= z^{\frac{2-\mu}{2}} \left( t + \frac{2(2a+\eta+1)}{(\eta-1)(\eta+3)} z^{\frac{\mu}{2}} \right) + \frac{a}{1-\eta} z, \end{aligned} \tag{16}$$

where we take

$$\left(\theta(t) > 0 \text{ and } (2\alpha + \eta - 1)\,\theta\_t(t) < 0,\tag{17}$$

and Eq. (15) becomes an equation for the function *ϑ*ð Þ*t* . Solving it, for *σCu* >0, we obtain:

$$\mathcal{A}(t) = \tilde{C}\_{\theta} \left( \frac{2\sigma a \mathcal{C}\_{u}}{c^{2}} \right)^{-\frac{\eta+3}{2(\eta+1)}} t\_{2} F\_{1} \left( \frac{1}{2}, \frac{\eta+3}{2(\eta+1)}; \frac{3}{2}; -\frac{(\eta+1)c^{2}}{2\sigma a \mathcal{C}\_{u}} t^{2} \right) + C\_{\theta}, \tag{18}$$

where *C*~*<sup>ϑ</sup>* and *C<sup>ϑ</sup>* are constants and <sup>2</sup>*F*<sup>1</sup> is the hypergeometric Gauss function. Further we obtain the solutions of initial Eqs. (3)–(4) in parametric form:

$$\begin{split} y(t) &= -\frac{\alpha(\eta+3)}{c(2\alpha+\eta-1)} \cdot \ln\left(\vartheta(t)\right) \\ v(t) &= \left(-\frac{\tilde{C}\_{\theta}(\eta+3)}{2(2\alpha+\eta-1)}\right)^{\frac{2}{1-\eta}} \left((\eta+1)t^{2} + \frac{2\sigma\alpha C\_{u}}{c^{2}}\right)^{-\frac{1}{\eta+1}} (\vartheta(t))^{\frac{2-\alpha}{2\alpha+\eta-1}} \\ u(t) &= C\_{u} \left(-\frac{\tilde{C}\_{\theta}(\eta+3)}{2(2\alpha+\eta-1)}\right)^{\frac{2}{1-\eta}} \left((\eta+1)t^{2} + \frac{2\sigma\alpha C\_{u}}{c^{2}}\right)^{-\frac{1}{\eta+1}} (\vartheta(t))^{\frac{m+2\alpha+2}{2\alpha+\eta-1}} \end{split} \tag{19}$$

where the constant *<sup>C</sup>*~*<sup>ϑ</sup>* is chosen so that 2ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>η</sup>* � <sup>1</sup> *<sup>C</sup>*<sup>~</sup> *<sup>ϑ</sup>* <sup>&</sup>lt;0, which is consistent with Eq. (17). Using the asymptotic representation of hypergeometric Gauss function as *t* ! �∞ [18], we can take:

$$|\mathbf{C}\_{\theta} > |\check{\mathbf{C}}\_{\theta}| \frac{\pi}{2\sqrt{\eta + 1}} \left(\frac{2\sigma a \mathbf{C}\_{u}}{c^{2}}\right)^{-\frac{1}{\eta + 1}} \frac{\Gamma\left(\frac{1}{\eta + 1}\right)}{\Gamma\left(\frac{\eta + 3}{2(\eta + 1)}\right)}\tag{20}$$

B. *η* 6¼ 1

**Figure 5.**

**6**

**Figure 3.**

**Figure 4.**

requires some conditions:

Let us return to Eq. (4) and rewrite it in terms of the variable *<sup>ξ</sup>* <sup>¼</sup> *<sup>e</sup>*�*cy*

*<sup>c</sup>*<sup>2</sup> *<sup>υ</sup>* <sup>þ</sup> *σαCu*

To integrate this equation, we use the Lie group method of infinitesimal transformations [20]. We find a group invariant of a second prolongation of oneparameter symmetry group vector of (12), and with its help, we transform Eq. (12) into an equation of the first order. It turns out that nontrivial symmetry group

> *<sup>β</sup>* <sup>¼</sup> ð Þ *<sup>α</sup>* � <sup>2</sup> ð Þ *<sup>α</sup>* <sup>þ</sup> *<sup>η</sup>* <sup>þ</sup> <sup>1</sup> *<sup>c</sup>*<sup>2</sup> *α η*ð Þ <sup>þ</sup> <sup>3</sup> <sup>2</sup>

*<sup>c</sup>*<sup>2</sup> *ξα <sup>υ</sup><sup>η</sup>* ¼ � *αβ<sup>b</sup>*

*ξ*2

*υξξ* � *αβ*

– 20 20 40 60 80 100

v y

– 20 20 40 60 80 100

**1; 1/4; 0; 1; 1/13; <sup>3</sup> 13 ;**

**2; 1/8; 0; 2; 1/13; <sup>5</sup> <sup>52</sup> ;**

Out[22]=

(a) (b)

*(a) u y* ð Þ*; c* ¼ *1; Cu* ¼ *9; σ* ¼ *1; b* ¼ *0:1. (b) u y* ð Þ*; c* ¼ *1; σ* ¼ *1; b* ¼ *0:1; α* ¼ *1; β* ¼ �*1=4; ν* ¼ *0.*

**3; 1/12; 0; 3; 1/13; <sup>1</sup> 3 13 ;** **1; 1/4; 0; 1; 1/13; <sup>3</sup>**

**2; 1/8; 0; 2; 1/13; <sup>5</sup>**

**3; 1/12; 0; 3; 1/13; <sup>1</sup>**

10 5 5 10 15 20

**1; 1/4; 0; 1; 1/13; <sup>3</sup>**

**2; 1/8; 0;**

**2; 1/13; <sup>5</sup>**

**3; 1/12; 0; 3; 1/13; <sup>1</sup>**

u y

0.005 0.010 0.015 0.020 0.025 **13 ;**

**<sup>52</sup> ;**

**3 13 ;**

> Cu 9 Cu 1;

v y

*Mathematical Theorems - Boundary Value Problems and Approximations*

– 0.10 – 0.08 – 0.06 – 0.04 – 0.02

– 5 5 10 15 20

0.02

*v y* ð Þ*; c* ¼ *1; Cu* ¼ �*9; σ* ¼ �*1; b* ¼ �*0:1.*

0.04

0.06

0.08

0.10

*v y* ð Þ*; c* ¼ *1; Cu* ¼ *9; σ* ¼ *1; b* ¼ *0:1.*

u y

0.005 0.010 0.015 0.020 0.025

> *αβb <sup>c</sup>*<sup>2</sup> <sup>¼</sup> 0,

*α* :

**13 ;**

**<sup>52</sup> ;**

**3 13 ;**

(13)

*<sup>c</sup>*<sup>2</sup> *:* (12)

in order for *y*, *v*, and *u* to be real. Then one can see that all functions in Eq. (19) are

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis*

**v t ; 0.4; 0.1;**

**u t ; 0.4; 0.1;**

**u t ; 3; 0.1;**

**u t ; 0.4; 1.3;**

**u t ; 3; 1.3; 10u t ; 0.4; 2; u t ; 3; 2;**

**10v t ; 0.01;**

**v t ; 1.1;**

**u t ; 1.1;**

**10 <sup>5</sup>**

**10 <sup>5</sup>**

**v t ; 2; 10u t ; 0.01;**

**u t ; 2;**

**102**

**102**

**10 <sup>2</sup>**

**v t ; 3; 0.1;**

**v t ; 0.4; 1.3;**

**v t ; 3; 1.3; 10v t ; 0.4; 2; v t ; 3; 2;**

**102**

**102**

**10 <sup>2</sup>**

continuous bounded ones for*t* ∈ ℜ and *v*, *u* are positive. Hence, one may try to biologically interpret the functions *v y* ð Þ and *u y* ð Þ, and this requires further investigation. In **Figure 6** one may see the different curves *v y* ð Þ for *η* ¼ 0*:*1 and different *α*. **Figure 7** demonstrates *v y* ð Þ and *u y* ð Þ for two values *η* : *η* ¼ 0*:*1 and *η* ¼ 0*:*01, see **Figure 7**. Further, for larger values of *α* and *η*, it seems more convenient to present the curves *y t*ð Þ, *v t*ð Þ, and *u t*ð Þto analyze them (see **Figures 8**–**10**). One can see from Eq. (13) that *β* ≷0 when *α*≷ 2, and the case of *β* ¼ 0 and *α* ¼ 2 is presented in **Figure 11**.

– 40 – 20 0 20 40

– 40 – 20 0 20 40 5 10 15 20 25 30 35 u t

– 40 – 20 20 40

*c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:4;* <sup>∣</sup>*C*<sup>~</sup> *<sup>ϑ</sup>*<sup>∣</sup> <sup>¼</sup> *1.*

2

4

6

8

10

12

*c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:4;* <sup>∣</sup>*C*<sup>~</sup> *<sup>ϑ</sup>*<sup>∣</sup> <sup>¼</sup> *1.*

*DOI: http://dx.doi.org/10.5772/intechopen.91214*

*c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:4;* <sup>∣</sup>*C*<sup>~</sup> *<sup>ϑ</sup>*<sup>∣</sup> <sup>¼</sup> *1.*

**Figure 9.** *v t*ð Þ*; σαCu*

**Figure 10.** *u t*ð Þ*; σαCu*

**Figure 11.** *v t*ð Þ*;u t*ð Þ*; <sup>α</sup>* <sup>¼</sup> *2; σαCu*

**9**

**Figure 8.** *y t*ð Þ*; σαCu c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:4;* <sup>∣</sup>*C*~*ϑ*<sup>∣</sup> <sup>¼</sup> *1.*

*Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis DOI: http://dx.doi.org/10.5772/intechopen.91214*

in order for *y*, *v*, and *u* to be real. Then one can see that all functions in Eq. (19) are continuous bounded ones for*t* ∈ ℜ and *v*, *u* are positive. Hence, one may try to biologically interpret the functions *v y* ð Þ and *u y* ð Þ, and this requires further investigation. In **Figure 6** one may see the different curves *v y* ð Þ for *η* ¼ 0*:*1 and different *α*. **Figure 7** demonstrates *v y* ð Þ and *u y* ð Þ for two values *η* : *η* ¼ 0*:*1 and *η* ¼ 0*:*01, see **Figure 7**. Further, for larger values of *α* and *η*, it seems more convenient to present the curves *y t*ð Þ, *v t*ð Þ, and *u t*ð Þto analyze them (see **Figures 8**–**10**). One can see from Eq. (13) that *β* ≷0 when *α*≷ 2, and the case of *β* ¼ 0 and *α* ¼ 2 is presented in **Figure 11**.

**Figure 11.** *v t*ð Þ*;u t*ð Þ*; <sup>α</sup>* <sup>¼</sup> *2; σαCu c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:4;* <sup>∣</sup>*C*<sup>~</sup> *<sup>ϑ</sup>*<sup>∣</sup> <sup>¼</sup> *1.*

0 2 4 6

v y ; u y

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>0</sup>

*c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:35;* <sup>∣</sup>*C*~*ϑ*<sup>∣</sup> <sup>¼</sup> *1.*

– 20 – 10 10 20

2

4

6 y t v y

*Mathematical Theorems - Boundary Value Problems and Approximations*

**0.4; 1; 2; 2.2;**

**v y ; 0.01; u y ; 0.01; v y ; 0.1; u y ; 0.1;**

> **0.4; 0.01; 4; 0.01; 0.4; 0.1; 4; 0.1; 0.4; 1.1; 4; 1.1; 0.4; 2; 4; 2;**

0.5 1.0 1.5 2.0 2.5 3.0 3.5

*c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:4;* <sup>∣</sup>*C*<sup>~</sup> *<sup>ϑ</sup>*<sup>∣</sup> <sup>¼</sup> *1.*

**Figure 6.** *v y* ð Þ*; <sup>η</sup>* <sup>¼</sup> *<sup>0</sup>:1; σαCu*

2

*v y* ð Þ*;u y* ð Þ*; <sup>α</sup>* <sup>¼</sup> *<sup>0</sup>:4; σαCu*

**Figure 7.**

**Figure 8.** *y t*ð Þ*; σαCu*

**8**

*c2* <sup>¼</sup> *2; c* <sup>¼</sup> *1; C<sup>ϑ</sup>* <sup>¼</sup> *<sup>1</sup>:4;* <sup>∣</sup>*C*~*ϑ*<sup>∣</sup> <sup>¼</sup> *1.*

4

6

8

10
