Numerical Solutions For Singularly Perturbed Nonlinear ... · they normally form a nonlinear...
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IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 15, Issue 1 Ser. I (Jan – Feb 2019), PP 35-49
www.iosrjournals.org
DOI: 10.9790/5728-1501013549www.iosrjournals.org35 | Page
Numerical Solutions For Singularly Perturbed Nonlinear
Reaction Diffusion Boundary Value Problems
Hakkı DURU && Baransel GÜNEŞ Corresponding Author:Hakkı DURU
Abstract:In this article, the boundary value problem for singularly perturbed nonlinear reaction diffussion
equations are treated. The exponentially fitted difference schemes on a uniform mesh which is accomplished by
the method of integral identities with the use of exponential basis functions and interpolating quadrature rules
with weight and remainder term in integral form are presented. The stability and convergence analysis of the
method is discussed. The fully discrete scheme is shown to be convergent of order 1 in independent variable,
independently of the perturbation parameter. Some numerical experiments have been carried out to validate the
predicted theory.
Keywords:Difference schemes, Differential equation, Singular perturbation, Uniform mesh.
2010 Mathematical Subject Classification: 65L10, 65L11, 65L12
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Date of Submission: 16-01-2019 Date of acceptance: 02-02-2019
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I. Introduction
In this study, the problem of nonlinear reaction-diffusion boundary value is investigated:
−𝜀2𝑢′′ + 𝑎 𝑥 𝑢 𝑥 = 𝑓(𝑥, 𝑢)(1)
𝑢 0 = 𝜅0 , 𝑢 𝑙 = 𝜅1(2)
Such problems emerge as mathematical models of the research object in many areas of science.
Reaction-diffusion equations are typical mathematical models in biology, physics and chemistry. Chemical
reaction models are widely used in biological systems, population dynamics, and nuclear reactivity. These
equations usually depend on various parameters such that temperature, catalyst, diffusion rate, etc. Moreover,
they normally form a nonlinear dissipative system coupled by reaction between different substances [13].
Such equations also appear in the model of population distributions. J.G. Skellam’s article on “Random
dispersal in theoretical populations” has been a turning point [22]. A series of observations made by J.G.
Skellam deeply affected his space ecology work. First, he set up a link between the theoretical biological species
as a definition of movement, the random walk, and the diffusion equation as a definition of the distribution of
the organism at the population density scale of species, and use the field data for the increase in the populations
of the musk rats in Central Europe the connection is acceptable. Second, in parallel to Fisher’s earlier
contribution to genetics, he has presented reaction-diffusion equations in a theoretical ecologically effective
manner by combining a common definition of distribution with population dynamics. Third, especially using
both the various assumptions of linear (Malthusian) and logistic population growth rate terms, one and two
dimensional habitat geometry, and interspace between the habitat and surrounding environmental regimes,
Skellam studied reaction-diffusion models for a population distribution in a limited living space [4].
Reaction-diffusion mechanisms have been used to explain model formation in developmental biology
and experimental chemical systems [18]. One of the ways to obtain oscillatory solutions in chemical system
models is reaction-diffusion equations [3].
In the biological sense, morphogenesis, which is called chemical substances that diffuse into a tissue by
entering the reaction, constitutes the main feature of morphogenesis. In this context, it is shown that the
reaction-diffusion problems are utilized in the stability of the situations arising due to the morphogenesis
suggested by Alan Turing [24].
Reaction and diffusion terms are used in synchronization methods due to the spatial binding of the
particles in the regime of periodic or chaotic oscillations depending on the dominant force of dynamic behavior.
In recent times, reaction diffusion systems have received much attention as a prototype for model
formation. Said models (facades, spirals, targets, hexagons, strips and scattering solitons) are used in a variety of
reaction-diffusion systems.It has been argued that the reaction diffusion processes are a basis for processes
dependent on morphogenesis in biology [9] and may even be related to animal pigmentation, skin pigmentation
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[14], [17]. Other applications of reaction diffusion equations include ecological invasions [10], outbreak spread
[16], tumor growth [5], [21], [7] and wound healing [20]. Another reason for the interest in reaction diffusion
systems is the possibility of an analytical treatment, although there are nonlinear partial differential equations
[6], [15], [8], [23], [11].
Singular perturbation problems include boundary layers that change rapidly in the solution function. In
this type of problem, the classical difference schemes are not stable since the derivative of the solution is infinite
at boundary layer. Also, the exact solution of such problems usually is not found. For this reason, numerical
algorithms are needed. In this study, difference scheme with exponential coefficients are presented for
singularly perturbed nonlinear reaction diffusion problems with boundary layer. In constructing these schemes,
interpolation quadrature rules which are the remainder term in integral form and contain weight function are
used. The approach ratio of the solutions of the difference problems is 𝑂(2).
II. Preliminary In this section, some basic definitions and basic theorems as without proof will be given . The
interpolating quadrature rules that are used in the construction of the difference scheme with their remainder
term integral form and containing the base function are given. In addition, the quadrature formulas and notations
needed to establish in a equidistance mesh are given. The required differential and integral inequalities and
difference alike will be given unspecified.
Lemma 1.Let 𝑝(𝑥)be a integrable function (weight function) and 𝜎 -a real parameter. Then, the following
interpolating quadratic formulas are true:
𝑓 𝑥 𝑝 𝑥 𝑑𝑥 = [ 𝑝 𝑥 𝑑𝑥]𝑏
𝑎
𝜎𝑓 𝑏 + 1 − 𝜎 𝑓 𝑎 𝑏
𝑎
+𝑓 𝑎; 𝑏 ∫ 𝑥 − 𝑥 𝜎 𝑝 𝑥 𝑑𝑥𝑏
𝑎+ 𝑅 𝑓 (3)
𝑅 𝑓 = 𝑑𝑥𝑝 𝑥 𝑓(𝑛)(𝜉)𝑏
𝑎
𝑏
𝑎
𝐾∗𝑛−1 𝑥, 𝜉 𝑑𝜉, 𝑛 = 1 𝑜𝑟 2,
𝐾𝑠 𝑥, 𝜉 = 𝑇𝑠 𝑥 − 𝜉 − 𝑏 − 𝑎 −1 𝑥 − 𝑎 𝑏 − 𝜉 𝑠 , 𝑠 = 0,1,
𝑥 𝜎 = σb + (1 − σ)a , 𝑓 𝑎, 𝑏 =𝑓 𝑏 −𝑓(𝑎)
𝑏−𝑎,
𝑇𝑠 𝜆 = 𝜆𝑠
𝑠!, 𝜆 ≥ 0, 𝑇𝑠 𝜆 = 0, 𝜆 < 0.
In some cases, second term in (1) formula may add the remainder term:
∫ 𝑓 𝑥 𝑝 𝑥 𝑑𝑥𝑏
𝑎= 𝑓
𝑎+𝑏
2 ∫ 𝑝 𝑥 𝑑𝑥
𝑏
𝑎+ 𝑅∗ 𝑓 (4)
𝑅∗ 𝑓 = 𝑑𝑥𝑝 𝑥 𝑓(𝑛)(𝜉)𝑏
𝑎
𝑏
𝑎
𝐾∗𝑛−1 𝑥, 𝜉 𝑑𝜉
+ 𝑛 − 1 𝑓 𝑎, 𝑏 (𝑥 −𝑎 + 𝑏
2
𝑏
𝑎
)𝑝 𝑥 𝑑𝑥, 𝑛 = 1 𝑜𝑟 2,
𝐾𝑛 𝑥, 𝜉 = 𝑇𝑛 𝑥 − 𝜉 − 𝑇𝑛 𝑎 + 𝑏
2− 𝜉
+ 𝑏 − 𝑎 −1 𝑏 − 𝜉 𝑆 𝑎 + 𝑏
2− 𝑥 , 𝑠 = 0, 1.
Moreover,
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∫ 𝑓 ′ 𝑥 𝑝 𝑥 𝑑𝑥 = 𝑓[𝑎; 𝑏]𝑏
𝑎 ∫ 𝑝 𝑥 𝑑𝑥𝑏
𝑎+ 𝑅 𝑓 (5)
𝑅 𝑓 = − 𝑑𝑥𝑝′ 𝑥 𝑏
𝑎
𝑓 𝑛 𝜉 𝐾𝑛−1 𝑥, 𝜉 𝑑𝜉𝑏
𝑎
, 𝑛 = 1, 2.
In this formula, we determinate the same 𝐾𝑠 𝑥, 𝜉 function in (3) and (4) formulas, where
𝐾0 𝑎, 𝜉 = 𝐾0 𝑏, 𝜉 = 0,
𝐾1 𝑎, 𝜉 = 𝐾1 𝑏, 𝜉 =𝐾1 𝑥, 𝑎 = 𝐾1 𝑥, 𝑏 = 0,
𝐾1 𝑥, 𝜉 = 𝐾1 𝜉, 𝑥 ,
𝑑
𝑑𝑥𝐾1 𝑥, 𝜉 = 𝐾0 𝜉, 𝑥 [2].
III. Asymptotic Estimations Forthe (1)-(2) nonlinear problem, from the Taylor expanded, we can write
𝑓 𝑥, 𝑢 = 𝑓 𝑥, 0 +𝜕𝑓(𝑥, 𝑢 )
𝜕𝑢𝑢
where 𝑢 = 𝛾𝑢, 0 < 𝛾 < 1, also we say 𝐹 𝑥 = 𝑓(𝑥, 0) and 𝑏 𝑥 =𝜕𝑓 (𝑥 ,𝑢 )
𝜕𝑢 . Then
−𝜀2𝑢′′ + 𝑎 𝑥 − 𝑏 𝑥 𝑢 = 𝐹 𝑥 .
If we get
𝐴 𝑥 = 𝑎 𝑥 − 𝑏(𝑥)
we have
−𝜀2𝑢′′ + 𝐴(𝑥)𝑢 = 𝐹 𝑥 (6)
where the following conditions valid:
𝐴 𝑥 = 𝑎 𝑥 −𝜕𝑓 𝑥, 𝑢
𝜕𝑢≥ 𝛼 > 0
𝐹 𝑥 = 𝑓(𝑥, 0) ≥ 0.
We give these lemmas for the asymptotic estimates.
Lemma 2. Let 𝑣(𝑥)be a function and satisfy the following conditions:
𝐿𝑣 = 𝐹(𝑥) ≥ 0
𝜅0 ≥ 0, 𝜅1 ≥ 0. Then 𝑣(𝑥) ≥ 0 [1].
Proof. To prove the lemma, assume that𝑣 𝑥0 = 0, 𝑣 𝑥1 = 0 and 𝑣 𝑥0 ≤ 0, for 𝑥 ∈ (𝑥0 , 𝑥1) ⊂
(0, 𝑙). Then
𝑣 ′′ 𝜉 =𝑣 𝑥0 − 2𝑣(
𝑥0+𝑥1
2) + 𝑣 𝑥1
(𝑥1−𝑥0
2)2
≥ 0.
From this, it mean that 𝐿𝑣 < 0 and 𝑣 𝑥 < 0 for 𝑥 ∈ (𝑥0 , 𝑥1). This is contradictory to the hypothesis.
So lemma is true.∎
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Lemma 3.For a 𝑣 𝑥 ∈ ∁[0, 𝑙] ∩ ∁2(0, 𝑙) function, the following estimation is true:
𝑣(𝑥) ≤ 𝑣 0 + 𝑣 𝑙 + 𝛼−1 𝑚𝑎𝑥0 ≤ 𝑠 ≤ 𝑙
𝐿𝑣 𝑠 , 0 ≤ 𝑥 ≤ 𝑙(7)
[1].
Proof. Fort he proof of lemma, we consider the following barrier function
𝜓 𝑥 = ±𝑣 𝑥 + 𝑣 0 + 𝑣 𝑙 + 𝛼−1 𝑚𝑎𝑥0 ≤ 𝑠 ≤ 𝑙
𝐿𝑣 𝑠 , 0 ≤ 𝑥 ≤ 𝑙.
For the boundary conditions, from this following relations
𝜓 0 = ±𝑣 0 + 𝑣 0 + 𝑣 𝑙 + 𝛼−1 𝑚𝑎𝑥0 ≤ 𝑠 ≤ 𝑙
𝐿𝑣 𝑠 , 0 ≤ 𝑥 ≤ 𝑙
we get 𝜓 0 ≥ 0 and from this following
𝜓 𝑙 = ±𝑣 𝑙 + 𝑣 0 + 𝑣 𝑙 + 𝛼−1 𝑚𝑎𝑥0 ≤ 𝑠 ≤ 𝑙
𝐿𝑣 𝑠 , 0 ≤ 𝑥 ≤ 𝑙
𝜓 𝑙 ≥ 0 . Thus 𝐿𝜓 𝑥 ≥ 0. From here 𝜓 𝑥 ≥ 0. From the relation (1), the inequality (7) is valid.∎
Lemma 4.For the solution of the (1)-(2) nonlinear problem, the following estimations are true:
𝑢(𝑥) ≤ 𝐶, 0 < 𝑥 < 𝑙 ; 𝐴 𝑥 , 𝐹 𝑥 ∈ ∁[0, 𝑙](8)
𝑢′ (𝑥) ≤ 𝐶 1 +1
𝜀 𝑒
− 𝛼𝑥
𝜀 + 𝑒− 𝛼 𝑙−𝑥
𝜀 , 0 < 𝑥 < 𝑙, 𝐴 𝑥 , 𝐹 𝑥 ∈ ∁[0, 𝑙](9)
[1].
Proof. First, we Show that 𝑢(𝑥) ≤ 𝐶. From the relation (7)
𝑢(𝑥) ≤ 𝜅0 + 𝜅1 + 𝛼−1 𝑚𝑎𝑥0 ≤ 𝑠 ≤ 𝑙
𝐹 𝑠 ,
then u(x) ≤ C. Now we Show that the estimation (9) is true. Taking into account the relation (8), if we modify
the differential equation, we have the following estimation:
𝑢′′ (𝑥) = −1
𝜀2 𝐹 𝑥 − 𝐴 𝑥 𝑢 𝑥 ≤
𝐶
𝜀2 , 0 < 𝑥 < 𝑙(10)
Then, we needs to estimate for the values 𝑢′ (0) and 𝑢′ (𝑙) . Using the following formula:
𝑔′ 𝑥 = 𝑔 𝛼0; 𝛼1 − 𝐾0 𝜉, 𝑥 𝑔′′ 𝜉 𝑑𝜉
𝛼1
𝛼0
, 𝑔 ∈ ∁2 , 𝛼0 ≤ 𝑥 ≤ 𝑙.
In this formula, if we get 𝑔 𝑥 = 𝑢(𝑥), 𝑥 = 0, 𝛼0 = 0, 𝛼1 = 𝜀, then
𝑢′ 0 = 𝑢 𝜀 −𝜅0
𝜀 ≤
𝐶
𝜀(11)
In the same way, if we get 𝑔 𝑥 = 𝑢(𝑥), 𝑥 = 𝑙, 𝛼0 = 𝑙 − 𝜀, 𝛼1 = 𝑙, then
𝑢′ 𝑙 = 𝜅1−𝑢 𝑙−𝜀
𝜀 ≤
𝐶
𝜀(12)
Here, if we derivate the differential equation (6), then
−𝜀2𝑢′′ + 𝐴 𝑥 𝑢 𝑥 = 𝐹 𝑥
−𝜀2𝑢′′′ + 𝐴′ 𝑥 𝑢 𝑥 + 𝐴 𝑥 𝑢′ 𝑥 = 𝐹′ 𝑥 . We get
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𝑤 𝑥 = 𝑢′ 𝑥
and
𝜑 𝑥 = 𝐹′ 𝑥 − 𝐴′ 𝑥 𝑢 𝑥 + 𝐴 𝑥 𝑢′ 𝑥 ,
then
𝐿𝑤 𝑥 = 𝛷 𝑥 (13)
Moreover, from (11) and (12),
𝑤 0 = 𝑂 1
𝜀 , 𝑤 𝑙 = 𝑂
1
𝜀 (14)
We search the solution of the linear problem (13)-(14) as the form 𝑤 𝑥 = 𝑤0 𝑥 + 𝑤1 𝑥 , where the
functions 𝑤0 𝑥 and 𝑤1 𝑥 respectively are the solutions of the following problems:
𝐿𝑤0 = 𝛷 𝑥 , 0 < 𝑥 < 𝑙
𝑤0 0 = 𝑤0 𝑙 = 0(15)
and
𝐿𝑤1 = 0, 0 < 𝑥 < 𝑙
𝑤1 0 = 𝑂 1
𝜀 , 𝑤1 𝑙 = 𝑂
1
𝜀 (16)
For the solutions of the problem (15) according to the boundary conditions, we can write the following
estimation:
𝑤0(𝑥) ≤ 𝛼−1 𝑚𝑎𝑥0 ≤ 𝑠 ≤ 𝑙
𝛷 𝑠 .
Therefore, if 𝛷 𝑥 is uniformly bounded by 𝜀, we have
𝑤0(𝑥) ≤ 𝐶, 0 < 𝑥 < 𝑙(17)
Now, if we apply the maximum principle to the problem (16), then we get
𝑤0(𝑥) ≤ Ʌ(𝑥)(18)
Here the function Ʌ(x) is the solution of the following problems:
−𝜀2Ʌ′′ + 𝛼Ʌ = 𝐹 𝑥 , 0 < 𝑥 < 𝑙
Ʌ 0 = 𝑤1(0) , Ʌ 𝑙 = 𝑤1(𝑙) (19)
The solution of the constant coefficient problem (19) is obvious:
Ʌ 𝑥 =1
sinh( 𝛼𝑙
𝜀) 𝑤1(0) sinh(
𝛼 𝑙 − 𝑥
𝜀+ 𝑤1 𝑙 sinh(
𝛼𝑥
𝜀) .
From here, it easy true that the following estimation:
Ʌ 𝑥 ≤𝐶
𝜀 𝑒−
𝛼𝑥
𝜀 + 𝑒− 𝛼(𝑙−𝑥)
𝜀 (20)
This show that lemma is true.∎
IV. Construction Of Difference Schemes
4.1. Establishing the Exponential Fitted Difference Schemes. In this section, a difference scheme is established
for the (1)-(2) non-linear reaction-diffusion problem is equidistant grid.
Let’s make the following notations for this. Here, the following equidistant discrete point set say the equidistant
grid:
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𝜔 = {𝑥𝑖 = 𝑖, 𝑖 = 1, . . . , 𝑁 − 1 , =1
𝑁}, 𝜛 = 𝜔 ∪ {0, 𝑙}
The points 𝑥𝑖 said the point of mesh. The functions defined on this grid are also called grid
functions. = 𝑥𝑖 − 𝑥𝑖−1has called meshsize. For the grid function 𝑢: 𝜔 → ℝ as 𝑢 𝑥𝑖 = 𝑢𝑖 , the
following notations respectively are called the forward difference derivative, backward
difference derivative, second difference derivative
𝑢𝑥 ,𝑖 =𝑢𝑖+1 − 𝑢𝑖
,
𝑢𝑥 ,𝑖 =𝑢𝑖 − 𝑢𝑖−1
,
𝑢𝑥 𝑥 ,𝑖 =1
𝑢𝑥 ,𝑖 − 𝑢𝑥 ,𝑖
[19]. We establish the difference scheme forthe differential equation (1) on the equidistant mesh ωh . This
scheme is the exponential fitted difference scheme. The difference scheme is established by using interpolation
quadrature rules with the remaining terms in integral form and containing the weight function. For this purpose,
we multiply both sides of the differential equation (1) with the base function 𝜑𝑖 , and integrate from 𝑥𝑖−1 to
𝑥𝑖+1. Therefore, we have the following equation:
𝜒𝑖−1−1 𝐿𝑢𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
= 𝜒𝑖−1−1 −𝜀2𝑢′′ + 𝑎 𝑥 𝑢 𝑥 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
= 𝜒𝑖−1−1 𝑓 𝑥, 𝑢 𝜑𝑖𝑑𝑥,
𝑥𝑖+1
𝑥𝑖−1
where 𝜑𝑖 is the basis function and 𝛾𝑖 = 𝑎𝑖
𝜀 , 𝜑𝑖
1 𝑥 and 𝜑𝑖 2 𝑥 are respectively the
solutions of the following problems:
𝜀2𝜑𝑖(1)′′ − 𝑎𝑖𝜑𝑖
1 = 0, 𝜑𝑖 1 𝑥𝑖 = 1, 𝜑𝑗
1 𝑥𝑖−1 = 0 (21)
and
𝜀2𝜑𝑖(2)′′ − 𝑎𝑖𝜑𝑖
2 = 0, 𝜑𝑖 2 𝑥𝑖 = 1, 𝜑𝑗
2 𝑥𝑖+1 = 0 (22)
The basis function 𝜑𝑖(𝑥) is as follows
𝜑𝑖 =
𝜑𝑖
1 (𝑥) =𝑠𝑖𝑛𝛾𝑖 𝑥 − 𝑥𝑖−1
𝑠𝑖𝑛𝛾𝑖, 𝑥 ∈ (𝑥𝑖−1 , 𝑥𝑖)
𝜑𝑖 1 (𝑥) =
𝑠𝑖𝑛𝛾𝑖 𝑥𝑖+1 − 𝑥
𝑠𝑖𝑛𝛾𝑖, 𝑥 ∈ (𝑥𝑖 , 𝑥𝑖+1)
.
The function 𝑥𝑖 is defined as follows
𝜒𝑖 = −1 𝜑𝑖𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
=2𝑡𝑎𝑛
𝛾𝑖
2
𝛾𝑖.
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For the −−1 ∫ 𝜀2𝑢′′ 𝜑𝑖𝑑𝑥𝑥𝑖+1
𝑥𝑖−1term, if partial integration rule is used, it is found the following
equation:
𝜒𝑖−1 −1𝜀2𝑢′𝜑𝑖|𝑥𝑖−1
𝑥𝑖+1 − −1𝜀2 𝑢′𝜑𝑖′𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
= 𝜒𝑖−1 −1𝜀2 𝑢′𝜑𝑖
1 ′𝑑𝑥𝑥𝑖
𝑥𝑖−1
+ 𝜀2 𝑢′𝜑𝑖 2 ′𝑑𝑥
𝑥𝑖+1
𝑥𝑖
If interpolating quadrature rule (5) is applied to this result, it is obtained the following result:
𝜒𝑖−1−1𝜀2 𝑢𝑥 ,𝑖 𝜑𝑖
1 ′𝑑𝑥𝑥𝑖
𝑥𝑖−1
+ 𝑑𝑥𝜑𝑖 1 ′′
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖
𝑥𝑖−1
𝑥𝑖
𝑥𝑖−1
𝑇1 𝑥 − 𝜉 𝑑𝜉
+ 𝑢𝑥 ,𝑗 𝜑𝑖 2 ′𝑑𝑥
𝑥𝑖+1
𝑥𝑖
+ 𝑑𝑥𝜑𝑖 2 ′′
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖+1
𝑥𝑖+1
𝑥𝑖+1
𝑥𝑖+1
𝑇1 𝑥 − 𝜉 𝑑𝜉
= −𝜒𝑖−1−1𝜀2 𝑢𝑥 ,𝑖 − 𝑢𝑥 ,𝑖 + 𝑅1,𝑖(𝑓)(23)
where 𝑅1,𝑖(𝑓)is defined as follows
𝑅 𝑓 = 𝜒𝑖−1 −𝜀2 𝑑𝑥𝜑𝑖
1 ′′𝑥𝑖
𝑥𝑖−1
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖
𝑥𝑖−1
𝑇1 𝑥 − 𝜉 𝑑𝜉
− 𝜀2 𝑑𝑥𝜑𝑖 2 ′′
𝑥𝑖+1
𝑥𝑖
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖+1
𝑥𝑖
𝑇1 𝑥 − 𝜉 𝑑𝜉 .
For the 𝜒𝑖−1−1 ∫ 𝑎 𝑥 𝑢(𝑥)𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1term, we obtain as follows
𝜒𝑖−1−1 𝑎 𝑥 𝑢 𝑥 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
= 𝜒𝑖−1−1 𝑎 𝑥 − 𝑎 𝑥𝑖 + 𝑎 𝑥𝑖 𝑢𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
= 𝜒𝑖−1−1 𝑎𝑖𝑢𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+ 𝑅∗𝑎 ,𝑖
where
𝑅∗𝑎 ,𝑖 = 𝜒𝑖
−1−1 𝑎 𝑥 − 𝑎 𝑥𝑖 𝑢𝜑𝑖𝑑𝑥.𝑥𝑖+1
𝑥𝑖−1
In addition to this result; if interpolation quadrature rule (4) is applied by taking as 𝜎 = 1
and𝑝 𝑥 = 𝜑𝑖 1 in interval of [𝑥𝑖−1, 𝑥𝑖],𝜎 = 0 and𝑝 𝑥 = 𝜑𝑖
2 in interval of [𝑥𝑖−1, 𝑥𝑖+1], the
following result is obtained:
𝜒𝑖−1−1[𝑎𝑖𝑢𝑖 𝜑𝑖
1 𝑑𝑥 + 𝑢𝑥 (𝑥 − 𝑥𝑖)𝜑𝑖 2 𝑑𝑥
𝑥𝑖+1
𝑥𝑖
𝑥𝑖
𝑥𝑖−1
+ 𝑑𝑥𝜑𝑖 2
𝑥𝑖+1
𝑥𝑖
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖+1
𝑥𝑖
𝑇1 𝑥 − 𝜉 𝑑𝜉
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+ 𝑑𝑥𝜑𝑖 1
𝑥𝑖
𝑥𝑖−1
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖
𝑥𝑖−1
𝑇1 𝑥 − 𝜉 𝑑𝜉 + 𝑢𝑖 𝜑𝑖 2 𝑑𝑥
𝑥𝑖+1
𝑥𝑖
]+𝑅∗𝑎 ,𝑖
= −𝜒𝑖−1−1{𝑎𝑖𝑢𝑖 𝜑𝑖𝑑𝑥 + 𝑎𝑖𝑢𝑥
𝑥𝑖+1
𝑥𝑖−1
𝑥 − 𝑥𝑖 𝜑𝑖 1 𝑑𝑥
𝑥𝑖
𝑥𝑖−1
+𝑎𝑖𝑢𝑥 𝑥 − 𝑥𝑖 𝜑𝑖 2 𝑑𝑥
𝑥𝑖+1
𝑥𝑖
+ 𝑎𝑖 𝑑𝑥𝜑𝑖 1
𝑥𝑖
𝑥𝑖−1
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖
𝑥𝑖−1
𝑇1 𝑥 − 𝜉 𝑑𝜉
+𝑎𝑖 ∫ 𝑑𝑥𝜑𝑖 2 𝑥𝑖+1
𝑥𝑖∫
𝑑2𝑢 𝜉
𝑑𝜉2
𝑥𝑖+1
𝑥𝑖𝑇1 𝑥 − 𝜉 𝑑𝜉}+𝑅∗
𝑎 ,𝑖(24)
If we combine the expressions of (23) and (24), and consider the expressions of (1) and (2), we have
the following expression
−𝜀2𝜃𝑖𝑢𝑥 𝑥 + 𝑎𝑖𝑢𝑖 + 𝑅∗𝑎 ,𝑖 = 𝜒𝑖
−1−1 ∫ 𝑓 𝑥, 𝑢 𝜑𝑖𝑑𝑥𝑥𝑖+1
𝑥𝑖−1(25)
where
𝜃𝑖 = 𝜒𝑖−1 1 + 𝜀−2𝑎𝑖 (𝑥 − 𝑥𝑖)𝜑𝑖
1 𝑑𝑥𝑥𝑖
𝑥𝑖−1
= 𝛾𝑖 2
𝑠𝑖𝑛𝛾𝑖
2
2 .
For the 𝜒𝑖−1−1 ∫ 𝑓 𝑥, 𝑢 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1 term, we obtain
𝜒𝑖−1−1 𝑓 𝑥, 𝑢 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
= 𝜒𝑖−1−1 { 𝑓 𝑥, 𝑢 − 𝑓 𝑥𝑖 , 𝑢 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+ 𝑓 𝑥𝑖 , 𝑢 − 𝑓 𝑥𝑖 , 𝑢𝑖 𝜑𝑖𝑑𝑥 + 𝑓 𝑥𝑖 , 𝑢𝑖 𝜒𝑖−1−1 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
}𝑥𝑖+1
𝑥𝑖−1
= 𝑓 𝑥𝑖 , 𝑢𝑖 + 𝑅𝑓 ,𝑖(26)
where the remainder term 𝑅𝑓 ,𝑖 is as follows
𝑅𝑓 ,𝑖 = 𝜒𝑖−1−1 𝑓 𝑥, 𝑢 − 𝑓 𝑥𝑖 , 𝑢 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+ 𝑓 𝑥𝑖 , 𝑢 − 𝑓 𝑥𝑖 , 𝑢𝑖 𝜑𝑖𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
If (21) and (22) are combined, following difference scheme is found:
−𝜀2𝜃𝑖𝑢𝑥 𝑥 ,𝑖 + 𝑎𝑖𝑢𝑖 = 𝑓 𝑥𝑖 , 𝑢𝑖 + 𝑅𝑖 , 𝑖 = 1, … , 𝑁 − 1
𝑢0 = 𝑢𝑁 = 0(27)
where the remainder term is 𝑅𝑖 = 𝑅𝑎 ,𝑖 + 𝑅𝑓 ,𝑖 .
For the approximating solution of 𝑦, we can write the following difference scheme
−𝜀2𝜃𝑖𝑢𝑥 𝑥 ,𝑖 + 𝑎𝑖𝑢𝑖 = 𝑓 𝑥𝑖 , 𝑢𝑖 , 𝑖 = 1, … , 𝑁 − 1
𝑦0 = 𝑦𝑁 = 0(28)
4.2. Error Estimations. . If we get 𝑧 = 𝑦 − 𝑢, we can write the following problem:
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−𝜀2𝜃𝑖𝑧𝑥 𝑥 ,𝑖 + 𝐴𝑖𝑧𝑖 = 𝑅𝑖 , 𝑖 = 1, … , 𝑁 − 1
𝑧0 = 𝑧𝑁 = 0.
where if the mean value theorem is applied to this problem, following expression is found:
𝑓 𝑥𝑖 , 𝑦𝑖 − 𝑓 𝑥𝑖 , 𝑢𝑖 =𝜕𝑓 𝑥𝑖 ,𝑢𝑖
𝜕𝑢 𝑦𝑖 − 𝑢𝑖 (29)
where
𝐴𝑖 = 𝑎𝑖 −𝜕𝑓 𝑥𝑖 ,𝑢𝑖
𝜕𝑢(30)
If we consider (29) and (30), error of 𝑧 satisfy the following boundary value difference problem:
𝑙𝑧𝑖 = 𝑅𝑖 , 𝑖 = 1, 𝑁 − 1, 𝑧0 = 𝑧𝑁 = 0 (31)
where the remainder term 𝑅𝑖 is determined as𝑅𝑖 = 𝑅𝑎 ,𝑖 + 𝑅𝑓 ,𝑖.
Theorem 5.For the 𝐴(𝑥) ∈ 𝐶1[0, 𝑙] , the solution of the difference problem (28) is uniformly
convergence to the solution of the problem (1)-(2) according to𝜀 in 𝜔 . Fort he error, the following
estimation is true:
𝑦 − 𝑢 ∁(𝜔 ) ≤ 𝐶
[1].
Proof. Where 𝜀 is arbitrary parameter and is meshsize, from the maximum principle for the
difference operator 𝑙𝑣𝑖 , if 𝑙𝑣𝑖 ≥ 0, (𝑖 = 1,2, … , 𝑁 − 1), 𝑣0 ≥ 0, 𝑣𝑁 ≥ 0then it can be shown that𝑣𝑖 ≥
0, 𝑖 = 0,1, … , 𝑁. If lemma 3 is applied to the problem (31), we can be written as follows
𝑧 ∁(𝜔 ) ≤ 𝛼−1 𝑅 ∁(𝜔 )(32)
Following estimation is obvious for 𝑅𝑖 , from 𝑅𝑎 ,𝑖 and 𝑅𝑓 ,𝑖 on the 𝐴(𝑥) ∈ 𝐶1[0, 𝑙],
𝑅𝑖 ≤ 𝐶 , 𝑖 = 1,2, … , 𝑁 − 1 (33)
Proof of theorem is completed from (32) and (33).∎
Remark 6. The difference problem (31) has only one solution. According to maximum principle, the
following linear system
𝑙𝑣𝑖 = 0, 𝑖 = 1, 𝑁 − 1, 𝑦0 = 𝑦𝑁 = 0 has only null solution. As known, this is a necessary and sufficient condition for the existence of a unique
solution for the system of linear equations [1].
Let’s consider the necessary conditions for convergence ratio to be 𝑂(2). We express this with a theorem.
Theorem 7. If 𝑎(𝑥) ∈ ∁2[0, 𝑙], 𝑓(𝑥, 𝑢) ∈ ∁2(𝐷)and
𝑎′ 0 = 𝑎′ 𝑙 = 0 (34)
the convergence ratio of the difference scheme (28) is 𝑂(2) and we can write as follows:
𝑦 − 𝑢 ∁(𝜔 ) ≤ 𝐶2 (35)
[1].
Proof. We consider the remainder term 𝑅𝑖 = 𝑅𝑎 ,𝑖 + 𝑅𝑓 ,𝑖, for proof of theorem where
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𝑅𝑎 ,𝑖 = −𝜒𝑖−1−1 [
𝑥𝑖+1
𝑥𝑖−1
𝑎 𝑥 − 𝑎 𝑥𝑖 𝑢(𝑥)𝜑𝑖(𝑥)𝑑𝑥
𝑅𝑓 ,𝑖 = 𝜒𝑖−1−1 𝑓 𝑥, 𝑢 − 𝑓 𝑥𝑖 , 𝑢 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+ 𝑓 𝑥𝑖 , 𝑢 − 𝑓 𝑥𝑖 , 𝑢𝑖 𝜑𝑖𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
.
First, we show that
𝑅𝑎 ,𝑖 = 𝑂 2 , 𝑖 = 1, 𝑁 − 1 (36)
If
𝑎 𝑥 − 𝑎 𝑥𝑖 = 𝑥 − 𝑥𝑖 𝑎′ 𝑥𝑖 +
𝑥 − 𝑥𝑖 2
2𝑎′′ 𝜉𝑖 , 𝜉𝑖 ∈ (𝑥𝑖 , 𝑥)
and
𝑢 𝑥 = 𝑢 𝑥𝑖 + 𝑥 − 𝑥𝑖 𝑢′ 𝜉𝑖 , 𝜉𝑖 ∈ (𝑥𝑖 , 𝑥)
equalities are written instead of 𝑅𝑎 ,𝑖, we obtain followings:
𝑅𝑎 ,𝑖 = −𝜒𝑖−1−1 𝑎′ 𝑥𝑖 𝑥 − 𝑥𝑖 𝑢 𝑥 𝜑𝑖𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
−1
2 𝑥 − 𝑥𝑖
2𝑎′′ 𝜉𝑖 𝑥 𝑢 𝑥 𝜑𝑖(𝑥)𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
= −𝜒𝑖−1−1 𝑎′ 𝑥𝑖 𝑢 𝑥𝑖 𝑥 − 𝑥𝑖 𝜑𝑖𝑑𝑥 + 𝑎′ 𝑥𝑖 𝑢′ 𝜉𝑖 𝑥 𝑥 − 𝑥𝑖
2𝜑𝑖 𝑥 𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
𝑥𝑖+1
𝑥𝑖−1
+1
2 𝑥 − 𝑥𝑖
2𝑎′′ 𝜉𝑖 𝑥 𝑢 𝑥 𝜑𝑖(𝑥)𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
= 𝜒𝑖−1−1𝑎′ 𝑥𝑖 𝑥 − 𝑥𝑖
2𝑢′ 𝜉𝑖 𝑥 𝜑𝑖 𝑥 𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
+1
2𝜒𝑖
−1−1 ∫ 𝑥 − 𝑥𝑖 2𝑎′′ 𝜉𝑖 𝑥 𝑢 𝑥 𝜑𝑖(𝑥)𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1 (37)
The second term on the right side of (37) satisfy the following inequality:
1
2𝜒𝑖
−1−1 ∫ 𝑥 − 𝑥𝑖 2𝑎′′ 𝜉𝑖 𝑥 𝑢 𝑥 𝜑𝑖(𝑥)𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1 ≤ 𝐶2, 𝑖 = 1,2, … , 𝑁 − 1 (38)
from 𝑎′′ (𝑥) ∁[0,𝑙] ≤ 𝐶, 𝑢 ∁[0,𝑙] ≤ 𝐶 and 𝑥 − 𝑥𝑖 2 ≤ 2. In order to evaluate the first term, if we
replace following inequality
𝑢′ 𝜉𝑖 ≤ 𝐶 1 +1
𝜀𝑒
− 𝛼𝜉𝑖𝜀 +
1
𝜀𝑒
− 𝛼(𝑙−𝜉𝑖)
𝜀
≤ 𝐶 1 +1
𝜀𝑒
− 𝛼𝑥𝑖−1𝜀 +
1
𝜀𝑒
− 𝛼 𝑙−𝑥𝑖+1
𝜀 , 1 < 𝑖 < 𝑁 − 1
with from the lemma 4, the first term in (37), we obtain
𝜒𝑖−1−1𝑎′ 𝑥𝑖 𝑥 − 𝑥𝑖
2𝑢′ 𝜉𝑖 𝑥 𝜑𝑖 𝑥 𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
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≤ 𝐶𝜒𝑖−1−1 𝑎′ 𝑥𝑖 𝑥 − 𝑥𝑖
2𝜑𝑖 𝑥 𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
+1
𝜀𝐶𝜒𝑖
−1−1 𝑎′ 𝑥𝑖 𝑥 − 𝑥𝑖 2𝜑𝑖 𝑥 𝑒
− 𝛼𝑥𝑖−1𝜀 𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+1
𝜀𝐶𝜒𝑖
−1−1 𝑎′ 𝑥𝑖 ∫ 𝑥 − 𝑥𝑖 2𝜑𝑖 𝑥 𝑒
− 𝛼 𝑙−𝑥𝑖+1
𝜀 𝑑𝑥𝑥𝑖+1
𝑥𝑖−1(39)
It can be easily that the convergence ratio of the first term on the right side of (39) is 𝑂 2 . The
followings are seen from 𝑎′ 0 = 0and𝑥𝑒−𝑥 < 𝑒−𝑥
2 , (𝑥 ≥ 0) to the second term:
1
𝜀𝐶𝜒𝑖
−1−1 𝑎′ 𝑥𝑖 𝑥 − 𝑥𝑖 2𝜑𝑖 𝑥 𝑒
− 𝛼𝑥𝑖−1𝜀 𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
≤1
𝜀𝐶𝜒𝑖
−1−1 𝑎′′ 𝜉𝑖 𝑥𝑖𝑒− 𝛼𝑥𝑖−1
𝜀 𝑥 − 𝑥𝑖 2𝜑𝑖 𝑥 𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
≤ 𝐶02𝑥𝑖
𝜀𝑒
− 𝛼𝑥𝑖−1𝜀 ≤ 𝐶02
𝑥𝑖
𝑥𝑖−1 𝛼
𝛼𝑥𝑖−1
𝜀𝑒
− 𝛼𝑥𝑖−1𝜀
≤ 𝐶12𝑖
𝑖 − 1𝑒
− 𝛼𝑥𝑖−12𝜀 ≤ 𝐶2 , 𝑖 > 1.
By the same way, the convergence ratio of the third term on the right side of (38) is 𝑂 2 with
𝑎′ 𝑙 = 0 (for 𝑖 < 𝑁 − 1). So the equality (37) is proved for 𝑖 = 2,3, … , 𝑁 − 2. For 𝑖 = 1 (by the
same way for 𝑖 = 𝑁 − 1), if we use
𝑎 𝑥 − 𝑎 𝑥1 = 𝑥 − 𝑥1 𝑎′ 𝑥1 + 𝑥 − 𝑥1 2
2𝑎′′ 𝜉1 , 𝜉1 ∈ (𝑥1 , 𝑥)
and
𝑢 𝑥 = 𝑢 𝑥0 + 𝑢′ 𝜉 𝑥
𝑥0
𝑑𝜉,
the following is obtained:
𝑅𝑎 ,1 = −𝜒1−1−1{𝑎′ 𝑥1 𝑥 − 𝑥1 𝑢′ 𝜉
𝑥
𝑥0
𝑑𝜉 𝜑1𝑑𝑥𝑥2
𝑥0
+1
2𝜒1
−1−1 ∫ 𝑥 − 𝑥1 2𝑎′′ 𝜉1 𝑥 𝑢 𝑥 𝜑1(𝑥)𝑑𝑥𝑥2
𝑥0}(40)
The second term to the right of (40) is seen to be 𝑂 2 from (38). We can evaulate the first term from
𝑎′ 0 = 0and lemma 4 as follows:
𝜒1−1−1𝑎′ 𝑥1 𝑥 − 𝑥1 𝑢′ 𝜉
𝑥
𝑥0
𝑑𝜉 𝜑1𝑑𝑥𝑥2
𝑥0
≤ 𝑎′ 𝑥1 𝑢′(𝑥) 𝑑𝑥𝑥2
𝑥0
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≤ 𝐶𝑥1 𝑎′′ 𝜉1 1 +1
𝜀𝑒
− 𝛼𝑥𝑖−1𝜀 +
1
𝜀𝑒
− 𝛼 𝑙−𝑥𝑖+1
𝜀 𝑑𝑥𝑥2
𝑥0
≤ 𝐶12 +1
𝜀 𝑒
− 𝛼𝑥
𝜀 𝑑𝑥𝑥2
𝑥0
≤ 𝐶12 + 𝛼 −1 1 − 𝑒−2 𝛼
𝜀 = 𝑂 2
Thus we prove 𝑅(1)𝑎 ,𝑖 = 𝑂 2 , 𝑅(𝑁−1)
𝑎 ,𝑖 = 𝑂 2 . So (40) is satisfied.
Now we consider term 𝑅𝑓 ,𝑖 , 𝑢𝑚 =𝑚𝑖𝑛[0, 𝑙]
𝑢(𝑥) , 𝑢𝑀 =𝑚𝑎𝑥[0, 𝑙] 𝑢(𝑥) , 𝑥, 𝑢 ∈ 𝐷 = 0, 𝑙 × [𝑢𝑚 , 𝑢𝑀] ,
𝑓 𝑥, 𝑢 ∈ ∁2(𝐷), it follows
𝑅𝑓 ,𝑖 = 𝜒𝑖−1−1{ 𝑓 𝑥, 𝑢 𝑥 − 𝑓 𝑥𝑖 , 𝑢 𝑥 𝜑𝑖 𝑥 𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+ 𝑓 𝑥𝑖 , 𝑢 𝑥 − 𝑓 𝑥𝑖 , 𝑢 𝑥𝑖 𝜑𝑖(𝑥)𝑑𝑥𝑥𝑖+1
𝑥𝑖−1
}
Using closed derivative formula and Taylor expansion, we have
𝑓 𝑥, 𝑢 𝑥 − 𝑓 𝑥𝑖 , 𝑢 𝑥 = 𝑥 − 𝑥𝑖 𝜕𝑓 𝑥𝑖 , 𝑢𝑖
𝜕𝑥+
𝜕𝑓 𝑥𝑖 , 𝑢𝑖
𝜕𝑢
𝜕𝑢 𝑥𝑖
𝜕𝑥
+ 𝑥 − 𝑥𝑖
2
2! 𝜕2𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑥2+ 2
𝜕2𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑥𝜕𝑢
𝑑𝑢 𝜉𝑖
𝑑𝑥+
𝜕2𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑢2 𝑑𝑢 𝜉𝑖
𝑑𝑥
2
+𝜕𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑢2
𝑑2𝑢 𝜉𝑖
𝑑𝑥2 .
From here
𝑅𝑓 ,𝑖 = 𝜒𝑖−1−1 𝑥 − 𝑥𝑖
𝜕𝑓 𝑥𝑖 , 𝑢𝑖
𝜕𝑥+
𝜕𝑓 𝑥𝑖 , 𝑢𝑖
𝜕𝑢
𝜕𝑢 𝑥𝑖
𝜕𝑥 𝜑𝑖 𝑥 𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+ 𝑥 − 𝑥𝑖
2
2! 𝜕2𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑥2+ 2
𝜕2𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑥𝜕𝑢
𝑑𝑢 𝜉𝑖
𝑑𝑥
𝑥𝑖+1
𝑥𝑖−1
+𝜕2𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑢2 𝑑𝑢 𝜉𝑖
𝑑𝑥
2
+𝜕𝑓 𝜉𝑖 , 𝑢 𝜉𝑖
𝜕𝑢2
𝑑2𝑢 𝜉𝑖
𝑑𝑥2 𝜑𝑖 𝑥 𝑑𝑥,
𝑅𝑓 ,𝑖 = 𝑂 2 (41)
is obvious from lemma 4, first and second derivative estimation for u function. Finally it is seen
𝑅𝑖 ∁(𝜔 ) ≤ 𝐶2
from (36) and (41) in the following equality
𝑅𝑖 = 𝑅𝑎 ,𝑖 + 𝑅𝑓 ,𝑖,
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Theorem is proved from this and (32).
V. Numerical Example The difference schemes in this section tested on the following non-linear problem:
−𝜀2𝑢′′ 𝑡 + 𝑥(1 − 𝑥)𝑢 𝑡 = 𝑢 + 𝑢2 , 𝑥 ∈ (0,1)
𝑢 0 = 𝑢 1 = 0(42)
Let’s write this problem explicitly for the (27) and (28):
−𝜀2−2𝜃𝑖 𝑦𝑖+1 𝑛 − 2𝑦𝑖
𝑛 + 𝑦𝑖−1 𝑛 + 𝑎𝑖𝑦𝑖
𝑛 − 𝑦𝑖 𝑛 𝜕𝑓(𝑥𝑖 , 𝑦𝑖
𝑛−1 )
𝜕𝑢
= 𝑓 𝑥𝑖 , 𝑦𝑖 𝑛−1 − 𝑦𝑖
𝑛−1 𝜕𝑓 𝑥𝑖 , 𝑦𝑖 𝑛−1
𝜕𝑢.
We edit this according to following:
𝐴𝑖𝑦𝑖−1 − 𝐶𝑖𝑦𝑖 + 𝐵𝑖𝑦𝑖+1 = −𝐹𝑖 , 𝑖 = 1, … , 𝑁 − 1
𝑦0 = 𝑦𝑁 = 0
Where the initial iteration is𝑦𝑖 0 = −2, 𝑖 = 1, … , 𝑁 − 1. Also
𝐴𝑖 = 𝐵𝑖 = 𝜀2−2𝜃𝑖 ,
𝐶𝑖 = 2𝜀²𝜃𝑖𝜃𝑖 + 𝑎𝑖 −𝜕𝑓 𝑥𝑖 , 𝑦𝑖
𝑛−1
𝜕𝑢,
𝐹𝑖 = 𝑓 𝑥𝑖 , 𝑦𝑖 𝑛−1 − 𝑦𝑖
𝑛−1 𝜕𝑓 𝑥𝑖 , 𝑦𝑖 𝑛−1
𝜕𝑢.
The elimination method and iteration should be applied together for the sample. 𝐶𝑖 − 𝛼𝑖𝐴𝑖 ≠ 0and the
elimination method is defined by
𝛼𝑖+1 =𝐵𝑖
𝐶𝑖 − 𝛼𝑖𝐴𝑖 , 𝛼₁ = 0, 𝑖 = 1, … , 𝑁 − 1
𝛽𝑖+1 =𝐹𝑖 + 𝐴𝑖𝛽𝑖
𝐶𝑖 − 𝛼𝑖𝐴𝑖 , 𝛽₁ = 0, 𝑖 = 1, … , 𝑁 − 1
and
𝑦𝑖 = 𝑦𝑖+1𝛼𝑖+1 + 𝛽𝑖+1 , 𝑖 = 𝑁 − 1, . . . ,0
[19].
Absolute errors are determined as
𝑟0 =𝑚𝑎𝑥
0 < 𝑖 < 𝑁 𝑦𝑖
− 𝑦𝑖
2
, 𝑟1 =𝑚𝑎𝑥
0 < 𝑖 < 𝑁 𝑦𝑖
2
− 𝑦𝑖
4
because the analytical solution of the test problem is not known. The smooth convergence ratio is calculated as
follows:
𝑝 =𝑙𝑛
𝑟1
𝑟2
𝑙𝑛2
[12]. Results are presented in following tables.Results in the solution of (42) test problem are calculated on
uniform mesh. Errors and p convergence ratios are given on Table (1)-(2).
Numerical Solutions For Singularly Perturbed Nonlinear Reaction Diffusion Boundary....
DOI: 10.9790/5728-1501013549www.iosrjournals.org48 | Page
Table 1. Convergence ratio at uniform mesh points for𝜀 = 2−𝑤 (𝑤 = 3, 4, . . . , 9)
𝜀 N=8 N=16 N=32 N=64
2−3 r0=0.311360
r1=0.042791 p=2.863215
r0=0.046374
r1=0.009127 p=2.345066
r0=0.009127
r1=0.002110 p=2.112825
r0=0.002110
r1=0.002110 p= 2.030424
2−4 r0=0.228723
r1= 0.181690 p=0.332122
r0=0.302153
r1=0.042560 p= 2.827703
r0=0.045168
r1=0.009008 p=2.326072
r0=0.009008
r1=0.002093 p=2.105377
2−5 r0=0.029511
r1=0.021707 p=0.443117
r0=0.213193
r1=0.173294 p=0.298938
r0=0.293582
r1=0.042189 p=2.798826
r0=0.044029
r1=0.008846 p=2.315451
2−6 r0=0.013264
r1=0.003572 p=1.892868
r0=0.034541
r1=0.019164 p=0.849931
r0=0.202879
r1=0.167515 p=0.276334
r0=0.287519
r1=0.041926 p=2.777732
2−7 r0=0.001015
r1=0.000229 p=2.145248
r0=0.017297
r1=0.003867 p=2.161150
r0=0.037469
r1=0.017750 p=1.077877
r0=0.197143
r1=0.164258 p=0.263277
2−8 r0=0.000009 r1=0.000015
p=-0.780445
r0=0.002291 r1=0.000280
p=3.032668
r0=0.020158 r1=0.004050
p=2.315307
r0=0.039032 r1=0.017014
p=1.197961
2−9 r0=0.000000 r1=0.000001
p=-5.619925
r0=0.000048 r1=0.000008
p=2.529604
r0=0.003855 r1=0.000335
p=3.526566
r0=0.021867 r1=0.004149
p=2.397955
Table 2. Convergence ratio at uniform mesh points for 𝜀 = 2−𝑤 (𝑤 = 3, 4, . . . , 9)
𝜀 N=128 N=256 N=512 N=1024
2−3 r0=0.000518 r1=0.000129
p=2.008042
r0=0.000129 r1=0.000032
p=2.002020
r0=0.000032 r1=0.000008
p=2.000501
r0=0.000008 r1=0.000002
p= 2.000123
2−4 r0=0.002103
r1= 0.000515
p=2.030544
r0=0.000516
r1=0.000128
p= 2.007474
r0=0.000128
r1=0.000032
p=2.001877
r0=0.000032
r1=0.000008
p=2.105377
2−5 r0=0.008854 r1=0.002061
p=2.102758
r0=0.002081 r1=0.000510
p=2.029223
r0=0.000510 r1=0.000127
p=2.007438
r0=0.00012 r1=0.000032
p=2.001832
2−6 r0=0.043257 r1=0.008739
p=2.307372
r0=0.008806 r1=0.002041
p=2.109253
r0=0.002068 r1=0.000507
p=2.0282252
r0=0.000507 r1=0.000126
p=2.007188
2−7 r0=0.284069 r1=0.041776
p=2.76549
r0=0.042822 r1=0.008680
p=2.302650
r0=0.008779 r1=0.002030
p=2.112542
r0=0.002060 r1=0.000505
p=2.027695
2−8 r0=0.194140 r1=0.162543
p=0.256272
r0=0.282245 r1=0.041696
p=2.758954
r0=0.042593 r1=0.008648
p=2.300122
r0=0.008765 r1=0.002028
p=2.111332
2−9 r0=0.039837 r1=0.016639
p=1.259558
r0=0.192606 r1=0.161665
p=0.252646
r0=0.281309 r1=0.041655
p=2.755580
r0=0.42476 r1=0.008632
p=2.298817
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Hakkı DURU. "Numerical Solutions For Singularly Perturbed Nonlinear Reaction Diffusion
Boundary Value Problems." IOSR Journal of Mathematics (IOSR-JM) 15.1 (2019): 35-49.