# Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term

Authors: Essam R. El-Zahar
Journal: International Journal of Differential Equations
 (2016)

---

## Abstract

A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.


---

## Body

## 1. Introduction

Many mathematical problems that model real-life phenomena cannot be solved completely by analytical means. Some of the most important mathematical problems arising in applied mathematics are singular perturbation problems. These problems commonly occur in many branches of applied mathematics such as transition points in quantum mechanics, edge layers in solid mechanics, boundary layers in fluid mechanics, skin layers in electrical applications, and shock layers in fluid and solid mechanics. The numerical treatment of these problems is accompanied by major computational difficulties due to the presence of sharp boundary and/or interior layers in the solution. Therefore, more efficient and simpler computational methods are required to solve these problems.For the past two decades, many numerical methods have appeared in the literature, which cover mostly second-order singular perturbation boundary value problems (SPBVPs) [1–3]. But only few authors have developed numerical methods for higher order SPBVPs (see, e.g., [4–10]). However, most of them have concentrated on problems with smooth data. In fact some authors have developed numerical methods for problems with discontinuous data which gives rise to an interior layer in the exact solution of the problem, in addition to the boundary layer at the outflow boundary point. Most notable among these methods are piecewise-uniform mesh finite difference method [11–14] and fitted mesh finite element method [15, 16] for third- and fourth-order SPBVPs with a discontinuous source term. The aim of this paper is to employ a semianalytical method which is Differential Transform Method (DTM) as an alternative to existing methods for solving high-order SPBVPs with a discontinuous source term.DTM is introduced by Zhou [17] in a study of electric circuits. This method is a formalized modified version of Taylor series method where the derivatives are evaluated through recurrence relations and not symbolically as the traditional Taylor series method. The method has been used effectively to obtain highly accurate solutions for large classes of linear and nonlinear problems (see, e.g., [17–21]). There is no need for discretization, perturbations, and further large computational work and round-off errors are avoided. Additionally, DTM does not generate secular terms (noise terms) and does not need analytical integrations as other semianalytical methods like HPM, HAM, ADM, or VIM and so DTM is an attractive tool for solving differential equations.In this paper, a reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and DTM. First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical method for treating high-order singular perturbation problems with a discontinuous source term.


## 2. Differential Transform Method for ODE System

Let us describe the DTM for solving the following system of ODEs:(1)u1′t=f1t,u1,u2,…,un,u2′t=f2t,u1,u2,…,un,⋮un′t=fnt,u1,u2,…,un,subject to the initial conditions(2)uit0=ci,i=1,2,…,n.Let [t0,T] be the interval over which we want to find the solution of (1)-(2). In actual applications of the DTM, the Nth-order approximate solution of (1)-(2) can be expressed by the finite series(3)uit=∑k=0NUikt-t0k,t∈t0,T,i=1,2,…,n,where(4)Uik=1k!dkuitdtkt=t0,i=1,2,…,n,which implies that ∑k=N+1∞Ui(k)(t-t0)k is negligibly small. Using some fundamental properties of DTM (Table 1), the ODE system (1)-(2) can be transformed into the following recurrence relations:(5)Uik+1=Fik,U1,U2,…,Unk+1,Yi0=ci,i=1,2,…,n,where Fi(k,U1,U2,…,Un) is the differential transform of the function fi(t,u1,u2,…,un), for i=1,2,…,n. Solving the recurrence relation (5), the differential transform Ui(k), k>0, can be easily obtained.Table 1
Some fundamental operations of DTM.
Original function Transformed function u ( t ) = β v ( t ) ± w ( t ) U ( k ) = β V ( k ) ± β W ( k )  u ( t ) = v ( t ) w ( t ) U ( k ) =  ∑ l = 0 k V ( l ) W ( k - l )  u ( t ) = d m v ( t ) d t m U ( k ) = ( k + m ) ! k ! V ( k + m )  u ( t ) = β + t m U ( k ) = H [ m , k ] m ! k ! m - k ! β + t 0 m - k, H[m,k]=1,ifm-k≥00,ifm-k<0  u ( t ) = e λ t U ( k ) = λ k k ! e λ t 0  u ( t ) = sin ⁡ ω t + β U ( k ) = ω k k ! sin ⁡ ω t 0 + β + k π 2  u ( t ) = cos ⁡ ω t + β U ( k ) = ω k k ! cos ⁡ ω t 0 + β + k π 2


## 3. Description of the Method

Motivated by the works of [11, 13, 16], we, in the present paper, suggest an asymptotic semianalytic method which is DTM to develop piecewise approximate analytical solutions for the following class of SPBVPs.Third-Order SPBVP [13]. Find y∈C1(Ω-)∩C2(Ω)∩C3Ω-∪Ω+ such that (6)-εy′′′t+aty′′t+bty′t+ctyt=ht,t∈Ω-∪Ω+,y0=p,y′0=q,y′1=r,where a(t), b(t), and c(t) are sufficiently smooth functions on Ω- satisfying the following conditions:(7)at<0,bt≥0,0≥ct≥-γ,γ>0,α-θγ≥η>0,where θ  is  arbitrarily  close  to  1, for some η.Fourth-Order SPBVP [11]. Find y∈C2(Ω-)∩C3(Ω)∩C4Ω-∪Ω+ such that (8)-εyivt+aty′′′t+bty′′t-ctyt=-ht,t∈Ω-∪Ω+,y0=p,y1=q,y′′0=-r,y′′1=-s,where a(t), b(t), and c(t) are sufficiently smooth functions on Ω- satisfying the following conditions:(9)at<0,bt≥0,0≥ct≥-γ,γ>0,α-θγ≥η>0,where θis  arbitrarily  close  to  1,  for  someη. For both the problems defined above, Ω=(0,1), Ω-=(0,d), Ω+=(d,1), Ω-=Ω-∪Ω+, Ω-=Ω∖{d}, and 0<ε≪1. It is assumed that h(t) is sufficiently smooth on Ω- and its derivatives have discontinuity at the point d and the jump at d is given as h(d)=h(d+)-h(d-).


### 3.1. Zero-Order Asymptotic Expansion Approximations

The SPBVP (6) can be transformed into an equivalent problem of the form(10)y1′t-y2t=0,t∈Ω∪1,-εy2′′t+aty2′t+bty2t+cty1t=ht,t∈Ω-,y10=p,y20=q,y21=r,where y1∈C1(Ω-) and y2∈C0(Ω-)∩C1(Ω)∩C2Ω-∪Ω+ [14].Similarly the SPBVP (8) can be transformed into(11)-y1′′t-y2t=0,t∈Ω∪1,-εy2′′t+aty2′t+bty2t+cty1t=ht,t∈Ω-,y10=p,y11=q,y20=r,y21=s,where y1∈C2(Ω-)∩C3(Ω)∩C4Ω-∪Ω+ and y2∈C0(Ω-)∩C1(Ω)∩C2Ω-∪Ω+ [11].Remark 1.
Hereafter, only the above systems (10) and (11) are considered.Using some standard perturbation methods [11, 13, 16, 22] one can construct an asymptotic expansion for the solution of (10) and (11) as follows.Find a continuous functionu=(u1,u2)T of the terminal value reduced system of (10) such that(12)u1′t-u2t=0,atu2′t+btu2t+ctu1t=ht,u10=p,u1d-=u1d+,u2d-=u2d+,u21=r.That is, find a smooth function u1 on Ω- that satisfies the following equivalent reduced BVP:(13)atu1′′t+btu1′t+ctu1t=ht,t∈Ω-,u10=p,u1d-=u1d+,u1′d-=u1′d+,u1′1=r.Then find(14)u2t=u1′t.Define yas=(y1,as,y2,as)T on Ω- as(15)y1,ast=u1t+εa0q-u20eta0/ε+kt,t∈Ω-∪0,d,kεadet-dad/ε,t∈Ω+∪1,y2,ast=u2t+q-u20eta0/ε+k,t∈Ω-∪0,d,ket-dad/ε,t∈Ω+∪1,where k=-εu2′(d+)-u2′(d-)/a(d).Similarly one can construct an asymptotic expansion for the solution of (11). In fact, for this problem u=(u1,u2)T is the solution of the terminal value reduced system (16)-u1′′t-u2t=0,atu2′t+btu2t+ctu1t=ht,t∈Ω-∪Ω+,u10=p,u1d-=u1d+,u1′d-=u1′d+,u2d-=u2d+,u11=q,u21=s.That is, in particular, find a smooth function u1 on Ω- that satisfies the following equivalent reduced BVP:(17)atu1′′′t+btu1′′t-ctu1t=-ht,t∈Ω-∪Ω+,u10=p,u1d-=u1d+,u1′d-=u1′d+,u1′′d-=u1′′d+,u11=q,u1′′1=-s.Then find u2(t)=-u1′′(t).Defineyas=(y1,as,y2,as)T on Ω- as(18)
y
1
,
a
s
t
=
u
1
t
+
-
k
1
ε
2
a
0
2
e
a
0
t
/
ε
,
t
∈
Ω
-
∪
0
,
d
,
-
k
2
ε
2
a
d
2
e
t
-
d
a
d
/
ε
,
t
∈
Ω
+
∪
1
,
y
2
,
a
s
t
=
u
2
t
+
k
1
e
a
0
t
/
ε
,
t
∈
Ω
-
∪
0
,
d
,
k
2
e
t
-
d
a
d
/
ε
,
t
∈
Ω
+
∪
1
,where(19)k1=r-u20,k2=-εa0u2′d+-u2′d-+k1ea0d/ε.Theorem 2 (see [11, 13]).
The zero-order asymptotic expansionyas=(y1,as,y2,as)T defined above for the solution y=(y1,y2)T of (10) and (11) satisfies the inequality(20)y-yas≤Cε.Now, in order to obtain piecewise analytical solutions of (10) and (11), we only need to obtain piecewise analytical solutions of the terminal value reduced systems (12) and (16), that is, the solution of equivalent reduced BVPs (13) and (17).


### 3.2. Piecewise Approximate Analytical Solutions

The solutionu1(t) of BVP (13) can be represented as a piecewise solution form:(21)u1t=u1Lt,t∈Ω-u1Rt,t∈Ω+.Thus the BVP (13) is transformed into(22)atu1L′′t+btu1L′t+ctu1Lt=ht,u1L0=p,u1L′0=α1,t∈Ω-,atu1R′′t+btu1R′t+ctu1Rt=ht,u1R1=β1,u1R′1=r,t∈Ω+,with continuity and smoothness conditions u1L(d-)=u1R(d+), u1L′(d-)=u1R′(d+) and α1, β1 are unknown constants.ApplyingNth-order DTM on (22) results in the recurrence relations(23)∑l=0kAk-ll+2l+1U1Ll+2+∑l=0kBk-ll+1U1Ll+1+∑l=0kCk-lU1Ll=Hk,U1L0=p,U1L1=α1,∑l=0kAk-ll+2l+1U1Rl+2+∑l=0kBk-ll+1U1Rl+1+∑l=0kCk-lU1Rl=Hk,U1R0=β1,U1R1=r,where A(k), B(k), C(k), H(k), U1L(k), and U1R(k) are the differential transform of a(t), b(t), c(t), h(t), u1L(t), and u1R(t), respectively, and α1 and β1 values are determined from the transformed continuity and smoothness conditions:(24)∑k=0NU1Lkd-k=∑k=0NU1Rkd+-1k,∑k=0Nk+1U1Lk+1d-k=∑k=0Nk+1U1Rk+1d+-1k.The recurrence relations (23) with transformed conditions (24) represent a system of algebraic equations in the coefficients of the power series solution of the reduced BVP (13) and the unknowns α1 and β1. Solving this algebraic system, the piecewise smooth approximate solution u~=(u~1(t),u~2(t))T of (13) is obtained and given by(25)
u
~
1
t
=
∑
k
=
0
N
U
1
L
k
t
k
,
t
∈
Ω
-
∑
k
=
0
N
U
1
R
k
t
-
1
k
,
t
∈
Ω
+
,
u
~
2
t
=
∑
k
=
0
N
k
+
1
U
1
L
k
+
1
t
k
,
t
∈
Ω
-
∑
k
=
0
N
k
+
1
U
1
R
k
+
1
t
-
1
k
,
t
∈
Ω
+
.And thus, the piecewise approximate analytical solution yap=(y1,ap,y2,ap)T of (10) is obtained and given by(26)y1,apt=u~1t+εa0q-u~20eta0/ε+kεt,t∈0,d,κε2adet-dad/ε,t∈d,1,y2,apt=u~2t+q-u~20eta0/ε+kε,t∈0,d,kεet-dad/ε,t∈d,1,where k=u~2′(d-)-u~2′(d+)/a(d).Similarly the reduced BVP (17) can be transformed into(27)atu1L′′′t+btu1L′′t-ctu1Lt=-ht,u1L0=p,u1L′0=α1,u1L′′0=α2,t∈Ω-,atu1R′′′t+btu1R′′t-ctu1Rt=-ht,u1R1=q,u1R′1=β1,u1R′′1=-s,t∈Ω+,with continuity and smoothness conditions u1L(d-)=u1R(d+), u1L′(d-)=u1R′(d+), u1L′′(d-)=u1R′′(d+) and α1, α2, β1 are unknown constants.ApplyingNth-order DTM on (27) results in the recurrence relations(28)∑l=0kAk-ll+3l+2l+1U1Ll+3+∑l=0kBk-ll+2l+1U1Ll+2-∑l=0kCk-lU1Ll=-Hk,U1L0=p,U1L1=α1,2U1L2=α2,∑l=0kAk-ll+3l+2l+1U1Rl+3+∑l=0kBk-ll+2l+1U1Rl+2-∑l=0kCk-lU1Rl=-Hk,U1R0=q,U1R1=β1,2U1R2=-s,where the unknown constants α1, α2, and β1 are determined from the transformed continuity and smoothness conditions:(29)∑k=0NU1Lkd-k=∑k=0NU1Rkd+-1k,∑k=0Nk+1U1Lk+1d-k=∑k=0Nk+1U1Rk+1d+-1k,∑k=0Nk+2k+1U1Lk+2d-k=∑k=0Nk+2k+1U1Rk+2d+-1k.And the piecewise approximate analytical solution yap=(y1,ap,y2,ap)T of (11) is obtained and given by(30)
y
1
,
a
p
t
=
u
~
1
t
+
k
1
ε
2
a
0
2
e
a
0
t
/
ε
,
t
∈
0
,
d
,
k
2
ε
2
a
d
2
e
t
-
d
a
d
/
ε
,
t
∈
d
,
1
,
y
2
,
a
p
t
=
u
~
2
t
+
k
1
e
a
0
t
/
ε
,
t
∈
0
,
d
,
k
2
e
t
-
d
a
d
/
ε
,
t
∈
d
,
1
,where(31)k1=r-u~20,k2=-εa0u~2′d--u~2′d++r-u~20ea0d/ε.


### 3.3. Error Estimate

The error estimate of the present method has two sources: one from the asymptotic approximation and the other from the truncated series approximation by DTM.Theorem 3.
Lety=(y1,y2)T be the solution of (10). Further let yap=(y1,ap,y2,ap)T be the approximate solution (26). Then(32)y-yap≤Cε+1N+1!.Proof.
Since the DTM is a formalized modified version of the Taylor series method, then we have a bounded error given by(33)u-u~≤MN+1!,M≤uN+1ξ,0≤ξ≤1.From Theorem 2 and the above bounded error, we have(34)y-yap≤y-yas+yas-yap≤C1ε+MN+1!.Since the singular perturbation parameter ε is extremely small, the present method works well for singular perturbation problems.Remark 4.
A similar statement is true for the solution of (11) and the approximate solution (30).


## 3.1. Zero-Order Asymptotic Expansion Approximations

The SPBVP (6) can be transformed into an equivalent problem of the form(10)y1′t-y2t=0,t∈Ω∪1,-εy2′′t+aty2′t+bty2t+cty1t=ht,t∈Ω-,y10=p,y20=q,y21=r,where y1∈C1(Ω-) and y2∈C0(Ω-)∩C1(Ω)∩C2Ω-∪Ω+ [14].Similarly the SPBVP (8) can be transformed into(11)-y1′′t-y2t=0,t∈Ω∪1,-εy2′′t+aty2′t+bty2t+cty1t=ht,t∈Ω-,y10=p,y11=q,y20=r,y21=s,where y1∈C2(Ω-)∩C3(Ω)∩C4Ω-∪Ω+ and y2∈C0(Ω-)∩C1(Ω)∩C2Ω-∪Ω+ [11].Remark 1.
Hereafter, only the above systems (10) and (11) are considered.Using some standard perturbation methods [11, 13, 16, 22] one can construct an asymptotic expansion for the solution of (10) and (11) as follows.Find a continuous functionu=(u1,u2)T of the terminal value reduced system of (10) such that(12)u1′t-u2t=0,atu2′t+btu2t+ctu1t=ht,u10=p,u1d-=u1d+,u2d-=u2d+,u21=r.That is, find a smooth function u1 on Ω- that satisfies the following equivalent reduced BVP:(13)atu1′′t+btu1′t+ctu1t=ht,t∈Ω-,u10=p,u1d-=u1d+,u1′d-=u1′d+,u1′1=r.Then find(14)u2t=u1′t.Define yas=(y1,as,y2,as)T on Ω- as(15)y1,ast=u1t+εa0q-u20eta0/ε+kt,t∈Ω-∪0,d,kεadet-dad/ε,t∈Ω+∪1,y2,ast=u2t+q-u20eta0/ε+k,t∈Ω-∪0,d,ket-dad/ε,t∈Ω+∪1,where k=-εu2′(d+)-u2′(d-)/a(d).Similarly one can construct an asymptotic expansion for the solution of (11). In fact, for this problem u=(u1,u2)T is the solution of the terminal value reduced system (16)-u1′′t-u2t=0,atu2′t+btu2t+ctu1t=ht,t∈Ω-∪Ω+,u10=p,u1d-=u1d+,u1′d-=u1′d+,u2d-=u2d+,u11=q,u21=s.That is, in particular, find a smooth function u1 on Ω- that satisfies the following equivalent reduced BVP:(17)atu1′′′t+btu1′′t-ctu1t=-ht,t∈Ω-∪Ω+,u10=p,u1d-=u1d+,u1′d-=u1′d+,u1′′d-=u1′′d+,u11=q,u1′′1=-s.Then find u2(t)=-u1′′(t).Defineyas=(y1,as,y2,as)T on Ω- as(18)
y
1
,
a
s
t
=
u
1
t
+
-
k
1
ε
2
a
0
2
e
a
0
t
/
ε
,
t
∈
Ω
-
∪
0
,
d
,
-
k
2
ε
2
a
d
2
e
t
-
d
a
d
/
ε
,
t
∈
Ω
+
∪
1
,
y
2
,
a
s
t
=
u
2
t
+
k
1
e
a
0
t
/
ε
,
t
∈
Ω
-
∪
0
,
d
,
k
2
e
t
-
d
a
d
/
ε
,
t
∈
Ω
+
∪
1
,where(19)k1=r-u20,k2=-εa0u2′d+-u2′d-+k1ea0d/ε.Theorem 2 (see [11, 13]).
The zero-order asymptotic expansionyas=(y1,as,y2,as)T defined above for the solution y=(y1,y2)T of (10) and (11) satisfies the inequality(20)y-yas≤Cε.Now, in order to obtain piecewise analytical solutions of (10) and (11), we only need to obtain piecewise analytical solutions of the terminal value reduced systems (12) and (16), that is, the solution of equivalent reduced BVPs (13) and (17).


## 3.2. Piecewise Approximate Analytical Solutions

The solutionu1(t) of BVP (13) can be represented as a piecewise solution form:(21)u1t=u1Lt,t∈Ω-u1Rt,t∈Ω+.Thus the BVP (13) is transformed into(22)atu1L′′t+btu1L′t+ctu1Lt=ht,u1L0=p,u1L′0=α1,t∈Ω-,atu1R′′t+btu1R′t+ctu1Rt=ht,u1R1=β1,u1R′1=r,t∈Ω+,with continuity and smoothness conditions u1L(d-)=u1R(d+), u1L′(d-)=u1R′(d+) and α1, β1 are unknown constants.ApplyingNth-order DTM on (22) results in the recurrence relations(23)∑l=0kAk-ll+2l+1U1Ll+2+∑l=0kBk-ll+1U1Ll+1+∑l=0kCk-lU1Ll=Hk,U1L0=p,U1L1=α1,∑l=0kAk-ll+2l+1U1Rl+2+∑l=0kBk-ll+1U1Rl+1+∑l=0kCk-lU1Rl=Hk,U1R0=β1,U1R1=r,where A(k), B(k), C(k), H(k), U1L(k), and U1R(k) are the differential transform of a(t), b(t), c(t), h(t), u1L(t), and u1R(t), respectively, and α1 and β1 values are determined from the transformed continuity and smoothness conditions:(24)∑k=0NU1Lkd-k=∑k=0NU1Rkd+-1k,∑k=0Nk+1U1Lk+1d-k=∑k=0Nk+1U1Rk+1d+-1k.The recurrence relations (23) with transformed conditions (24) represent a system of algebraic equations in the coefficients of the power series solution of the reduced BVP (13) and the unknowns α1 and β1. Solving this algebraic system, the piecewise smooth approximate solution u~=(u~1(t),u~2(t))T of (13) is obtained and given by(25)
u
~
1
t
=
∑
k
=
0
N
U
1
L
k
t
k
,
t
∈
Ω
-
∑
k
=
0
N
U
1
R
k
t
-
1
k
,
t
∈
Ω
+
,
u
~
2
t
=
∑
k
=
0
N
k
+
1
U
1
L
k
+
1
t
k
,
t
∈
Ω
-
∑
k
=
0
N
k
+
1
U
1
R
k
+
1
t
-
1
k
,
t
∈
Ω
+
.And thus, the piecewise approximate analytical solution yap=(y1,ap,y2,ap)T of (10) is obtained and given by(26)y1,apt=u~1t+εa0q-u~20eta0/ε+kεt,t∈0,d,κε2adet-dad/ε,t∈d,1,y2,apt=u~2t+q-u~20eta0/ε+kε,t∈0,d,kεet-dad/ε,t∈d,1,where k=u~2′(d-)-u~2′(d+)/a(d).Similarly the reduced BVP (17) can be transformed into(27)atu1L′′′t+btu1L′′t-ctu1Lt=-ht,u1L0=p,u1L′0=α1,u1L′′0=α2,t∈Ω-,atu1R′′′t+btu1R′′t-ctu1Rt=-ht,u1R1=q,u1R′1=β1,u1R′′1=-s,t∈Ω+,with continuity and smoothness conditions u1L(d-)=u1R(d+), u1L′(d-)=u1R′(d+), u1L′′(d-)=u1R′′(d+) and α1, α2, β1 are unknown constants.ApplyingNth-order DTM on (27) results in the recurrence relations(28)∑l=0kAk-ll+3l+2l+1U1Ll+3+∑l=0kBk-ll+2l+1U1Ll+2-∑l=0kCk-lU1Ll=-Hk,U1L0=p,U1L1=α1,2U1L2=α2,∑l=0kAk-ll+3l+2l+1U1Rl+3+∑l=0kBk-ll+2l+1U1Rl+2-∑l=0kCk-lU1Rl=-Hk,U1R0=q,U1R1=β1,2U1R2=-s,where the unknown constants α1, α2, and β1 are determined from the transformed continuity and smoothness conditions:(29)∑k=0NU1Lkd-k=∑k=0NU1Rkd+-1k,∑k=0Nk+1U1Lk+1d-k=∑k=0Nk+1U1Rk+1d+-1k,∑k=0Nk+2k+1U1Lk+2d-k=∑k=0Nk+2k+1U1Rk+2d+-1k.And the piecewise approximate analytical solution yap=(y1,ap,y2,ap)T of (11) is obtained and given by(30)
y
1
,
a
p
t
=
u
~
1
t
+
k
1
ε
2
a
0
2
e
a
0
t
/
ε
,
t
∈
0
,
d
,
k
2
ε
2
a
d
2
e
t
-
d
a
d
/
ε
,
t
∈
d
,
1
,
y
2
,
a
p
t
=
u
~
2
t
+
k
1
e
a
0
t
/
ε
,
t
∈
0
,
d
,
k
2
e
t
-
d
a
d
/
ε
,
t
∈
d
,
1
,where(31)k1=r-u~20,k2=-εa0u~2′d--u~2′d++r-u~20ea0d/ε.


## 3.3. Error Estimate

The error estimate of the present method has two sources: one from the asymptotic approximation and the other from the truncated series approximation by DTM.Theorem 3.
Lety=(y1,y2)T be the solution of (10). Further let yap=(y1,ap,y2,ap)T be the approximate solution (26). Then(32)y-yap≤Cε+1N+1!.Proof.
Since the DTM is a formalized modified version of the Taylor series method, then we have a bounded error given by(33)u-u~≤MN+1!,M≤uN+1ξ,0≤ξ≤1.From Theorem 2 and the above bounded error, we have(34)y-yap≤y-yas+yas-yap≤C1ε+MN+1!.Since the singular perturbation parameter ε is extremely small, the present method works well for singular perturbation problems.Remark 4.
A similar statement is true for the solution of (11) and the approximate solution (30).


## 4. Illustrating Examples

In this section we will apply the method described in the previous section to find piecewise approximate analytical solutions for three SPBVPs with a discontinuous source term.Example 1.
Consider the third-order SPBVP from [13, 16](35)-εy′′′t-2y′′t+4y′t-2yt=ht,y0=1,y′0=0,y′1=0,where(36)ht=0.7,0≤t≤0.5-0.6,0.5<t≤1.Using the present method with 5th-order DTM, the piecewise analytical solution is given by(37)y1,apt=1+0.575447t-0.099553t2-0.162277t3-0.072842t4-0.021023t5+0.287723εe-2t/ε+ε20.966783-0.849226et,t∈0,0.5,1.330163-0.515081t-12-0.343388t-13-0.128770t-14-0.034339t-15+0.287723εe-2t/ε-ε240.966783-0.849226ee-2t-0.5/ε,t∈0.5,1.0,y2,apt=0.575447-0.199107t-0.486830t2-0.291369t3-0.105115t4-0.575447e-2t/ε+ε20.966783-0.849226e,t∈0,0.5,-1.030163t+1.030163-1.030163t-12-0.515081t-13-0.171694t-14-0.575447e-2t/ε+ε20.966783-0.849226ee-2t-0.5/ε,t∈0.5,1.0.Example 2.
Consider the third-order SPBVP with variable coefficients from [13, 16](38)-εy′′′t-2ety′′t+cos⁡πt4y′t-1+xyt=ht,y0=0,y′0=0,y′1=1,where(39)ht=2t3,0≤t≤0.510t+1,0.5<t≤1.Using the present method with 5th-order DTM, the piecewise analytical solution is given by(40)y1,apt=1.865844t+0.466461t2-0.233230t3-0.072568t4+0.000470t5+0.932922εe-2t/ep-0.513641εt,t∈0,0.5,0.814311+t-1.280360t-12-0.069249t-13+0.181448t-14-0.052651t-15+0.932922εe-2t/ε+0.155770ε2e-3.297442t-0.5/ε,t∈0.5,1.0,y2,apt=1.865844+0.932922t-0.699692t2-0.290271t3+0.002350t4-1.865844e-2t/ε-0.513641εt,t∈0,0.5,3.560719-2.560719t-0.207747t-12+0.725794t-13-0.263254t-14-1.865844e-2t/ε-0.513641εe-3.297442t-0.5/ε,t∈0.5,1.0.Example 3.
Consider the fourth-order SPBVP from [13, 16] (41)-εyivt-4y′′′t+4y′′t=-ht,t∈Ω-∪Ω+,y0=1,y1=1,y′′0=-1,y′′1=-1,where(42)ht=0.7,0≤t≤0.5-0.6,0.5<t≤1.Using the present method with 5th-order DTM, the piecewise analytical solution is given by(43)y1,apt=1+0.256751t-0.199842t2-0.037447t3-0.009362t4-0.0018724t5-0.037520ε2e-4t/ε,t∈0,0.5,1.353655-0.353655t-12t-12-23120t-13-23480t-14-232400t-15-1162.5εE-11+0.600316e-2.0/εε2e-4t-0.5/ε,t∈0.5,1.0,y2,apt=0.399683+0.224683t+0.112342t2+0.037447t3+0.600316e-4t/ε,t∈0,0.5,-320+2320t+2340t-12+23120t-13+2.5εE-11+0.600316e-2.0/εe-4t-0.5/ε,t∈0,1.0.The numerical solution for each example is presented overall the problem domain as shown in Figures 1–3.
The corresponding maximum pointwise errors are taken to be(44)EεN=maxti∈Ω-εNyapNti-yap50ti,EN=max⁡EεN,where yapN(ti) is the obtained approximate solution using Nth-order DTM over a uniform mesh ti=ih, ti∈[0,1], h=10-3, i=0,1,2,…and yap50(ti) is our numerical reference solution obtained using DTM with order N=50.
The computed maximum pointwise errorsEεN and EN for the above solved BVPs are given in Tables 2–7. The numerical results in Tables 2–7 agree with the theoretical ones present in this paper where the obtained solutions and their derivatives converge rapidly to the reference solutions with increasing the order of the DTM.Table 2
Maximum pointwise errorsEεN and EN for the solution y1,ap of Example 1.
ε Approximation order of DTM,N 4 6 8 10 2 - 3 2.9574 e - 3 6.5563 e - 5 5.7278 e - 7 6.3740 e - 9 2 - 9 2.9573 e - 3 3.0803 e - 5 1.6200 e - 7 4.0000 e - 9 2 - 15 2.9573 e - 3 3.0803 e - 5 1.6200 e - 7 4.0000 e - 9 2 - 21 2.9573 e - 3 3.0803 e - 5 1.6200 e - 7 4.0000 e - 9  E  N 2.9574 e - 3 6.5563 e - 5 5.7278 e - 7 6.3740 e - 9Table 3
Maximum pointwise errorsEεN and EN for the first derivative solution y2,ap of Example 1.
ε Approximation order of DTM,N 4 6 8 10 2 - 3 8.8781 e - 3 1.6938 e - 4 1.3570 e - 6 8.6900 e - 9 2 - 9 1.0321 e - 2 1.3075 e - 4 7.8084 e - 7 2.3233 e - 9 2 - 15 1.0444 e - 2 1.3312 e - 4 8.0028 e - 7 2.6769 e - 9 2 - 21 1.0444 e - 2 1.3316 e - 4 8.0067 e - 7 2.4481 e - 9  E  N 1.0444 e - 2 1.6938 e - 4 8.0067 e - 7 8.6900 e - 9Table 4
Maximum pointwise errorsEεN and EN for the solution y1,ap of Example 2.
ε Approximation order of DTM,N 4 6 8 10 2 - 3 1.2547 e - 2 1.0369 e - 3 1.3608 e - 5 1.5365 e - 6 2 - 9 1.2547 e - 2 1.0077 e - 3 1.3488 e - 5 6.8490 e - 7 2 - 15 1.2547 e - 2 1.0077 e - 3 1.3488 e - 5 6.8100 e - 7 2 - 21 1.2547 e - 2 1.0077 e - 3 1.3488 e - 5 6.8100 e - 7  E  N 1.2547 e - 2 1.0369 e - 3 1.3488 e - 5 1.5365 e - 6Table 5
Maximum pointwise errorsEεN and EN for the first derivative solution y2,ap of Example 2.
ε Approximation order of DTM,N 4 6 8 10 2 - 3 1.8191 e - 2 2.2563 e - 3 2.9397 e - 5 3.2970 e - 6 2 - 9 2.1534 e - 2 2.0165 e - 3 2.6724 e - 5 1.5660 e - 6 2 - 15 2.1583 e - 2 2.0124 e - 3 2.6679 e - 5 1.5390 e - 6 2 - 21 2.1584 e - 2 2.0123 e - 3 2.6678 e - 5 1.5390 e - 6  E  N 2.1584 e - 2 2.2563 e - 3 2.9397 e - 5 3.2970 e - 6Table 6
Maximum pointwise errorsEεN and EN for the solution y1,ap of Example 3.
ε Approximation order of DTM,N 4 6 8 10 2 - 3 1.8918 e - 3 2.3001 e - 5 1.3700 e - 7 3.3000 e - 9 2 - 9 1.8918 e - 3 2.3001 e - 5 1.3700 e - 7 3.3000 e - 9 2 - 15 1.8918 e - 3 2.3001 e - 5 1.3700 e - 7 3.3000 e - 9 2 - 21 1.8918 e - 3 2.3001 e - 5 1.3700 e - 7 3.3000 e - 9  E  N 1.8918 e - 3 2.3001 e - 5 1.3700 e - 7 3.3000 e - 9Table 7
Maximum pointwise errorsEεN and EN for the second derivative solution y2,ap of Example 3.
ε Approximation order of DTM,N 4 6 8 10 2 - 3 2.1578 e - 2 2.8775 e - 4 1.7839 e - 6 6.5000 e - 9 2 - 9 2.1578 e - 2 2.8775 e - 4 1.7839 e - 6 6.5000 e - 9 2 - 15 2.1578 e - 2 2.8775 e - 4 1.7839 e - 6 6.5000 e - 9 2 - 21 2.1578 e - 2 2.8775 e - 4 1.7839 e - 6 6.5000 e - 9  E  N 2.1578 e - 2 2.8775 e - 4 1.7839 e - 6 6.5000 e - 9Figure 1
Graphs of the approximate solutiony1,ap and its first derivative y2,ap for Example 1 at ε=2-9 and N=5.
(a) (b)Figure 2
Graphs of the approximate solutiony1,ap and its first derivative y2,ap for Example 2 at ε=2-9 and N=5.
(a) (b)Figure 3
Graphs of the approximate solutiony1,ap and the second derivative y2,ap for Example 3 at ε=2-9 and N=5.
(a) (b)


## 5. Conclusion

We have presented a new reliable algorithm to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion SPBVPs with a discontinuous source term. The algorithm is based on constructing a zero-order asymptotic expansion of the solution and the DTM which provides the solutions in terms of convergent series with easily computable components. The original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion for the solution of the transformed system is constructed. For simplicity, the result terminal value reduced system is replaced by its equivalent reduced BVP with suitable continuity and smoothness conditions. Then a piecewise smooth solution of the reduced BVP is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented and shows that the method results in high-order convergence for small values of the singular perturbation parameter. We have applied the method on three SPBVPs and the piecewise analytical solution is presented for each one overall the problem domain. The numerical results confirm that the obtained solutions and their derivatives converge rapidly to the reference solutions with increasing the order of the DTM. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order SPBVPs with a discontinuous source term. The method is based on a straightforward procedure, suitable for engineers.



---

*Source: 1015634-2016-11-15.xml*