# On Solution of Integrodifferential Equation with Delay Parameter by Sinc Basis Functions

Authors: I. Najafi Khalilsaraye; K. Maleknejad
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We want to find a numerical solution for an integrodifferential equation with an integral boundary condition and delay parameter. This type of problems arises in mathematical physics, mechanics, population growth, and other fields of physics and mathematical chemistry. So, convergence of this approach is discussed by presenting a theorem which gives exponential type convergence rate and guarantees the accuracy of that. Finally, by some numerical examples, we show the efficiency and accuracy of this numerical method.


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## Body

## 1. Introduction

Discussing integrodifferential equations with integral boundary condition consisting of delay parameter is a worthy and significant branch of nonlinear applied mathematics. It is important that integrodifferential equation with delay parameter is generated often in investigations connected with chemical engineering, mathematical physics, underground water flow, engineering, and so on (see [1, 2]). Note that the problems with integral boundary conditions have various applications in applied fields such as population growth problems and blood flow problems. For a detailed description of the integrodifferential equations with delay parameter and problems with integral boundary conditions, the reader can refer to references of [3–7].In this paper we discuss the following problem which shows a first-order integrodifferential equation with integral boundary condition consisting of delay parameter. The existence and uniqueness of solution for this problem are proved in [8]. But analytic solving and reaching an exact solution are impossible. Then in this paper we approximate the exact solution by numerical method
(1)dydx=g(x,y(x),y(γ1(x)),Uy(x),Vy(x))≡Gy(x),y(0)=η1y(κ)+η2∫0b‍ω(r,y(r))dr+a,
where x∈I=[0,b] (b>0), g∈C(I×ℝ4,ℝ), where C is family of all continuous functions, and γ1∈C(I,I), κ∈(0,b], ω∈C(I×ℝ,ℝ),  η1,η2,a∈ℝ.And(2)(Uy)(x)=∫0γ2(x)‍k(x,r)y(B(r))dr,(Vy)(x)=∫0b‍h(x,r)y(D(r))dr.
Here γ2,B,D∈C(I,I), k(x,r)∈C[A,ℝ+], and h(x,r)∈C[A0,ℝ+] that
(3)A={(x,r)∈ℝ2∣0≤r≤γ2(x),x∈I},A0={(x,r)∈ℝ2∣0≤r≤x,x∈I}.
Here if η1=1, η2=a=0, and κ=b, then we have a problem with boundary condition of kind periodic, and if η1=0, then we have a problem with integral boundary condition, and if η1=η2=0, we have a problem with an initial condition. So, problem (1) is general type of these cases.Now, the following functional integral equation can be easily concluded from (1):
(4)y(x)=η1y(κ)+η2∫0b‍ω(r,y(r))dr+a+∫0xg(t,y(t),y(γ1(t)),∫0γ2(t)‍k(t,r)y(B(r))dr,‍∫0b‍h(t,r)y(D(r))dr)dt
and in this paper we want to discuss solution of (1) in point of equivalent integral equation (4). But we know the fact that we cannot solve this integral equation to give an exact solution, so numerical approaches are used to reach an approximated solution.Numerical approaches for estimated solution of integrodifferential and integral equation and search for existence and uniqueness of solution for some problems have been researched by many authors and reader can see these methods in [9–15]. In these references authors use methods on an estimate by basis function such as wavelets, polynomials, and so forth or use some quadrature formulas. But these technics usually have convergence rate of polynomial order with respect to M where M represents the cardinal of terms of sum in the expansion or the cardinal of points of the quadrature formula. In [16] author showed that if we employ the Sinc approach, the convergence rate is exponential order such as O(exp⁡(-cM1/2)) with some c>0. We know the exponential rate is much faster than that of polynomial rate. So, in this paper, we employ the Sinc function instead of base function and an iterative technique to estimate exact solution of (4) in marked points. Our approach dose not contain of changing the solution of (4) to a system of algebraic equations by expanding y(x) as basis function with unknown coefficients, so this technique has computations less than other methods and exponential rate in accuracy. Also in this present paper, we prove a theorem to guarantee the convergence of numerical technique.


## 2. Main Results

In this section, we introduce basic requirements and theorem to prove existence and uniqueness of solution for first-order integrodifferential equation with integral boundary condition consisting of delay parameter (1). For detailed descriptions, we refer the reader to [17, 18].Definition 1.
A function asy∈C1(I,I) is called a lower solution of (1) if
(5)dydx≤Gy(x),x∈I,y(0)≤η1y(κ)+η2∫0b‍ω(r,y(r))dr+a
and it is an upper solution of (1) if
(6)dydx≥Gy(x),x∈I,y(0)≥η1y(κ)+η2∫0b‍ω(r,y(r))dr+a.Theorem 2.
Letu0,v0∈C1(I,ℝ) be lower and upper solutions of (1), respectively, and u0(x)≤v0(x), x∈I.
In addition consider the following.(L1):
g∈C(I×ℝ4,ℝ), γ1,γ2,B,D∈C(I,I), γ1(x),γ2(x),B(x),D(x)≤x, ∀x∈I, κ∈(0,b), ω∈C(I×ℝ,ℝ), and η1,η2≥0.(L2):
there are nonnegative bounded integrable functionsM1(x), M2(x), M3(x), M4(x) on I that
(7)∫0b‍[M1(t)+M2(x)+M3(x)∫0γ2(x)‍k(x,r)dr+M4(x)∫0b‍h(x,r)dr]dx≤1
such that
(8)g(x,ϕ1,ϕ2,Uϕ1,Vϕ1)-g(x,ψ1,ψ2,Uψ1,Vψ1)≥-M1(x)(ϕ1-ψ1)-M2(x)(ϕ2-ψ2)-M3(x)U(ϕ1-ψ1)-M4(x)V(ϕ1-ψ1)
if u0≤ψ1≤ϕ1≤v0, u0(γ1(x))≤ψ2≤ϕ2≤v0(γ1(x)).(L3):
there isθ(x)∈C(I,ℝ+) such that ω(x,ψ)-ω(x,ψ-)≥θ(x)(ψ-ψ-), and if u0(t)≤ψ-≤ψ≤v0(x). then problem (1) has extremal solutions u,v∈[u0,v0]. In addition, there are monotone sequences un(x),vn(x)⊂[u0,v0] such that un→u, vn→v for n→+∞ and these are convergent uniformly on x∈I, where un(x), vn(x) are defined as
(9)un(x)=∫0x‍e-∫rx‍M1(s)ds×[g(r,un-1(r),un-1(γ1(r)),Uun-1(r),Vun-1(r)(γ1(r)))+M1(r)un-1(r)-M2(r)(un-un-1)(γ1(r))-M3(r)U(un-un-1)(r)-M4(r)V(un-un-1)(r)(γ1(r))]dr+e-∫0x‍M1(s)ds[∫0b‍η1un-1(κ)+η2∫0b‍ω(r,un-1(r))dr+a];∀x∈I,n=1,2,3,…,vn(x)=∫0x‍e-∫rx‍M1(s)ds×[g(r,vn-1(r),vn-1(γ1(r)),Uvn-1(r),Vvn-1(r)(γ1(r)))+M1(r)zn-1(r)-M2(r)(vn-vn-1)(γ2(r))-M3(r)U(vn-vn-1)(r)-M4(r)V(vn-vn-1)(r)(r,vn-1(r),vn-1(γ1(r)),]dr+e-∫0x‍M1(s)ds[∫0b‍η1vn-1(κ)+η2∫0b‍ω(r,vn-1(r))dr+a];∀x∈I,n=1,2,3,…,u0≤u1≤⋯≤un≤⋯≤u≤v≤⋯≤vn≤⋯≤v1≤v0.Proof.
See [18].Theorem 3.
Consider that assumptions of Theorem2 hold. Moreover, consider the following.(L4):
there are nonnegative bounded functionsα1(x), α2(x), α3(x), α4(x) on I, such that
(10)g(x,ϕ1,ϕ2,Uϕ1,Vϕ1)-g(x,ψ1,ψ2,Uψ1,Vψ1)≤α1(x)(ϕ1-ψ1)+α2(x)(ϕ2-ψ2)+α3(x)U(ϕ1-ψ1)+α4(x)V(ϕ1-ψ1)
if u0(x)≤ψ1≤ϕ1≤v0(x), u0(γ1(x))≤ψ2≤ϕ2≤v0(γ1(x)).(L5):
there isβ(x)∈𝒞(I,ℝ+) such that ω(x,ψ)-ω(x,ψ-)≤β(x)(ψ-ψ-)   if u0(t)≤ψ-≤ψ≤v0(x).Then problem (1) has a unique solution u-∈[u0,v0]. In addition, there are sequences un(x),vn(x)⊂[u0,v0] that these are monotone and un→u-, vn→u- for n→+∞. This convergence is uniformly on x∈I, where un(x), vn(x) are defined as (9) such that ∥un-u-∥c≤Ln∥v0-u0∥c, n∈ℕ, where
(11)L=η1+∫0b‍[∫0b‍η2β(x)+α1(x)+M1(x)+α2(x)+M2(x)+(α3(x)+M3(x))∫0γ2(x)‍K(x,r)dr+(α4(x)+M4(x))∫0b‍h(x,r)dr]dx<1.Proof.
See   [18].


## 3. Sinc Function

In this section, we recall the basis function and some of its applicabilities. In here, definition ofsinc(x) function is followed by
(12)Sinc(x)={sin(πx)πx;x≠0,1;x=0.Now, forh>0 and integer j, we define jth Sinc function with step size h by
(13)S(j,h)(x)=sin(π(x-jh)/h)π(x-jh)/h.


### 3.1. Sinc Estimation on[a,b]

Letx=φ(w) be a transformation that denotes a conformal transformation which transfers the simply connected domain A onto a strip region Ad such that
(14)φ((a,b))=(-∞,∞),lim⁡x→aφ(x)=-∞,lim⁡x→bφ(x)=∞.
In here ∂A is boundary of A and in order to have the Sinc estimation on (a,b) conformal transformation is applied as follows:
(15)φ(t)=ln⁡(t-ab-t).
This function transfers the eye-shaped complex region
(16){w=x+iy:|arg(w-ab-w)|<d≤π2}
onto Ad that it is a strip region:
(17)Ad={σ=α+βi:|β|<d<π2}.
The basis functions on finite interval (a,b) are given by
(18)S(j,h)∘φ(x)=sin(π(φ(x)-jh)/h)π(φ(x)-jh)/h,
and also, Sinc function for interpolation points xj=jh is given by
(19)S(j,h)(kh)=δjk(0)={1;j=k,0;j≠k.
Then, S(j,h)∘φ(x) shows behavior of Kronecker delta function on the network points
(20)xj=φ-1(jh)=a+bejh1+ejh.
The approximation of g(x) by interpolation and quadrature formulas for ∫ab‍g(x)dx is
(21)g(x)≈∑j=-MM‍g(xj)S(j,h)∘φ(x),∫ab‍g(x)dx≈h∑j=-MM‍g(xj)φ′(xj).Theorem 4.
Consider that, for a mapw=φ-1(ξ), the map g(φ-1(ξ)) satisfies(1)
g∈H1(Dd), for d>0,(2)
g decays exponentially on the real line such that
(22)|g(x)|≤αexp⁡⁡(-β|x|),∀x∈ℝ,α,β>0
with someα, β, and d. Then one has
(23)sup⁡a<x<b|g(x)-∑j=-MM‍g(φ-1(jh))S(j,h)∘φ(x)|≤CMexp⁡(-πdβM).
That there is some C and h is h=πd/βM.Proof.
See [16].Definition 5.
LetLα(A) be the set of all analytic functions g, for which there exists a constant C, such that
(24)|g(z)|≤C|eφ(z)|α(1+|eφ(z)|)2α;z∈A,0<α≤1.Theorem 6.
Letg/φ′∈Lα(A), with 0<α≤1 and 0<d≤π; also let h=πd/αM. Then there exists a constant C1, which is independent of M, such that
(25)|∫axj‍g(t)dt-h∑k=-MM‍δjk(-1)g(xj)φ′(xk)|≤C1e-πdαM,
where
(26)δjk(-1)=12+∫0j-k‍sin⁡(πt)πtdt,
and φ(x),xk are defined as above.Proof.
See   [16].


### 3.2. Sinc-Qudrature Method

In this section, for solving equation(27)y(x)=λ1y(κ)+η2∫0b‍ω(r,y(r))dr+a+∫0xg‍(∫0γ2(s)‍k(s,r)y(B(r))drs,y(s),y(γ1(s)),∫0γ2(s)‍k(s,r)y(B(r))dr,∫0b‍h(s,r)y(D(s))dr)ds,
we try to discrete integral equation by quadrature formula as
(28)∫ab‍g(s)ds=h∑j=-MM‍g(sj)φ′(sj)+O(exp⁡(-2πdMlog⁡⁡(2πdM/β)))∫as‍f(x)dx=h∑j=-MM‍g(xj)φ′(xj)ηh,j(s)+O(log⁡MMexp⁡(-πdMlog⁡⁡(πdM/β))),
where
(29)ηh,j(s)=12+1πri(πr-jhh);ri=∫0x‍sin(t)tdt
with xj=sj=(a+bejh)/(1+jh), j=-M,…,M and h=(1/M)log⁡(πdM/β) (see [16]).Now, by substituting quadrature formulas in the integral equation (4), we have
(30)yM(x)=η1y(κ)+η2h∑j=-MM‍ω(sj,y(sj))φ′(sj)+a+h∑j=-MM‍g(∑i=-MM‍sj,y(sj),y(α(sj)),h∑i=-MM‍K(sj,ri)y(B(ri))φ′(ri)ηh,i(t),h∑i=-MM‍h(sj,ri)y(D(ri))φ′(ri))×(φ′(sj))-1ηh,j(x).Now, forn=1,2,3,…,   let
(31)y1,M(x)=η1y(κ),yn+1,M(x)=η1yn,M(κ)+η2h∑j=-MM‍ω(sj,yn,M(sj))φ′(sj)+a+h∑j=-MM‍g(∑i=-MM‍sj,yn,M(sj),yn,M(α(sj)),h∑i=-MM‍a(sj,ri)yn,M(B(ri))φ′(ri)ηh,i(x),h∑i=-MM‍h(sj,ri)yn,M(δ(ri))φ′(ri))×(φ′(sj))-1ηh,j(x).


## 3.1. Sinc Estimation on[a,b]

Letx=φ(w) be a transformation that denotes a conformal transformation which transfers the simply connected domain A onto a strip region Ad such that
(14)φ((a,b))=(-∞,∞),lim⁡x→aφ(x)=-∞,lim⁡x→bφ(x)=∞.
In here ∂A is boundary of A and in order to have the Sinc estimation on (a,b) conformal transformation is applied as follows:
(15)φ(t)=ln⁡(t-ab-t).
This function transfers the eye-shaped complex region
(16){w=x+iy:|arg(w-ab-w)|<d≤π2}
onto Ad that it is a strip region:
(17)Ad={σ=α+βi:|β|<d<π2}.
The basis functions on finite interval (a,b) are given by
(18)S(j,h)∘φ(x)=sin(π(φ(x)-jh)/h)π(φ(x)-jh)/h,
and also, Sinc function for interpolation points xj=jh is given by
(19)S(j,h)(kh)=δjk(0)={1;j=k,0;j≠k.
Then, S(j,h)∘φ(x) shows behavior of Kronecker delta function on the network points
(20)xj=φ-1(jh)=a+bejh1+ejh.
The approximation of g(x) by interpolation and quadrature formulas for ∫ab‍g(x)dx is
(21)g(x)≈∑j=-MM‍g(xj)S(j,h)∘φ(x),∫ab‍g(x)dx≈h∑j=-MM‍g(xj)φ′(xj).Theorem 4.
Consider that, for a mapw=φ-1(ξ), the map g(φ-1(ξ)) satisfies(1)
g∈H1(Dd), for d>0,(2)
g decays exponentially on the real line such that
(22)|g(x)|≤αexp⁡⁡(-β|x|),∀x∈ℝ,α,β>0
with someα, β, and d. Then one has
(23)sup⁡a<x<b|g(x)-∑j=-MM‍g(φ-1(jh))S(j,h)∘φ(x)|≤CMexp⁡(-πdβM).
That there is some C and h is h=πd/βM.Proof.
See [16].Definition 5.
LetLα(A) be the set of all analytic functions g, for which there exists a constant C, such that
(24)|g(z)|≤C|eφ(z)|α(1+|eφ(z)|)2α;z∈A,0<α≤1.Theorem 6.
Letg/φ′∈Lα(A), with 0<α≤1 and 0<d≤π; also let h=πd/αM. Then there exists a constant C1, which is independent of M, such that
(25)|∫axj‍g(t)dt-h∑k=-MM‍δjk(-1)g(xj)φ′(xk)|≤C1e-πdαM,
where
(26)δjk(-1)=12+∫0j-k‍sin⁡(πt)πtdt,
and φ(x),xk are defined as above.Proof.
See   [16].


## 3.2. Sinc-Qudrature Method

In this section, for solving equation(27)y(x)=λ1y(κ)+η2∫0b‍ω(r,y(r))dr+a+∫0xg‍(∫0γ2(s)‍k(s,r)y(B(r))drs,y(s),y(γ1(s)),∫0γ2(s)‍k(s,r)y(B(r))dr,∫0b‍h(s,r)y(D(s))dr)ds,
we try to discrete integral equation by quadrature formula as
(28)∫ab‍g(s)ds=h∑j=-MM‍g(sj)φ′(sj)+O(exp⁡(-2πdMlog⁡⁡(2πdM/β)))∫as‍f(x)dx=h∑j=-MM‍g(xj)φ′(xj)ηh,j(s)+O(log⁡MMexp⁡(-πdMlog⁡⁡(πdM/β))),
where
(29)ηh,j(s)=12+1πri(πr-jhh);ri=∫0x‍sin(t)tdt
with xj=sj=(a+bejh)/(1+jh), j=-M,…,M and h=(1/M)log⁡(πdM/β) (see [16]).Now, by substituting quadrature formulas in the integral equation (4), we have
(30)yM(x)=η1y(κ)+η2h∑j=-MM‍ω(sj,y(sj))φ′(sj)+a+h∑j=-MM‍g(∑i=-MM‍sj,y(sj),y(α(sj)),h∑i=-MM‍K(sj,ri)y(B(ri))φ′(ri)ηh,i(t),h∑i=-MM‍h(sj,ri)y(D(ri))φ′(ri))×(φ′(sj))-1ηh,j(x).Now, forn=1,2,3,…,   let
(31)y1,M(x)=η1y(κ),yn+1,M(x)=η1yn,M(κ)+η2h∑j=-MM‍ω(sj,yn,M(sj))φ′(sj)+a+h∑j=-MM‍g(∑i=-MM‍sj,yn,M(sj),yn,M(α(sj)),h∑i=-MM‍a(sj,ri)yn,M(B(ri))φ′(ri)ηh,i(x),h∑i=-MM‍h(sj,ri)yn,M(δ(ri))φ′(ri))×(φ′(sj))-1ηh,j(x).


## 4. Convergence of Method

In this section, we present a theorem that shows a bound fory(x)-yn(x) with the real norm where y(x) is the exact solution of problem (4) and yn(x) is an estimation for y(x) by using Sinc function in interpolaion and quadrature formula. The result is shown as follows.Theorem 7.
Under the assumptions(L1)–(L5), iterative approximation approach (31) is convergent to exact solution if y1(x) is closed enough to the it and
(32)N2≤1-(η1+η2bN1)b(2+N3+N4),
where N1=sup⁡{β(x);x∈I}, N2=max⁡{αi(x);i=1,2,3,4,x∈I}, N3=sup⁡{K(x,r);x,r∈I}, and N4=sup⁡{h(x,r);x,r∈I}.Proof.
For a fixedM let
(33)yn+1,M(x)=η1xn,M(τ)+η2∫0b‍ω(r,yn,M(r))dr+a+∫0x‍g(s,yn,M(s),yn,M(γ1(s)),Uyn,M(s),Vyn,M(s))ds,y(x)=η1y(κ)+η2∫0b‍ω(r,y(r))dr+a+∫0x‍g(s,x(s),y(γ1(s)),Uy(s),Vy(s))ds.
Now
(34)|yn+1,M(x)-y(x)|≤η1|yn(κ)-y(κ)|+η2|∫0b‍(ω(r,yn(r))-ω(r,y(r)))dr|+|∫0x‍[g(∫0b‍s,yn(s),yn(γ1(s)),∫0γ2(s)‍K(s,r)yn(B(r))dr,∫0b‍h(s,r)yn(D(r))dr)-f(∫0b‍s,y(s),y(γ1(s)),∫0γ2(s)‍K(s,r)y(B(r))dr,∫0b‍h(s,r)y(D(r))dr)]ds|≤η1|yn(κ)-y(κ)|+η2bN1|yn(x)-y(x)|+bN2|yn(x)-y(x)|+bN2|yn(γ1(x))-y(γ1(x))|+bN2N3|xn(B(r))-y(B(r))|+bN2M4|yn(D(r))-y(D(r))|≤(η1+η2bN1+bN2(2+N3+N4))|yn,M(x)-y(x)|.
Then
(35)|yn+1,M(x)-y(x)|≤(η1+η2bN1+bN2(2+N3+N4))n×|y1,M(x)-y(x)|.
Now, by assumption in theorem, we have η1+η2bN1+bN2(2+N3+N4)<1.
Becauseη1+η2bN1+bN2(2+N3+N4)<η1+η2bN1+b((1-(η1+η2bN1))/b(2+N3+N4))(2+N3+   N4)=1 and then lim⁡n→+∞yn+1,M(x)=y(x), 0≤x≤b.


## 5. Illustrative Examples

In this section, for showing efficiency of the iterative scheme and in order to show the facts of the exact solution, we give some examples below. All routines have been written in Mathematica 7 and a Dual-Core CPU 2.00 GHz is used to run the programs. Also, about efficiency and accuracy of the proposed numerical method, we present absolute errors for different examples. Therefore numerical results are shown in figures to illustrate the efficiency of this scheme.Example 1.
Consider the following problem:(36)dydx=115x3[x-y(x)]-15xy(γ1(x))+1100x2y2(γ1(x))+1500x[x3-∫0(1/2)x‍8xry(r2)dr]5-1600x2[x3-∫0(1/2)x‍8xry(r2)dr]6+1700x2[x2-∫01‍2x2ry(r2)dr]7-1800x3[x2-∫01‍2x2ry(r2)dr]8≡Gy(x),-1800[zx-∫01‍2x2ry(r2)dr]8x∈I=[0,1],y(0)=14y(κ)+14∫01‍(2r+y(r))dr+14,
where κ∈(0,1], γ1∈C(I,I), and γ1(x)≤x on I.Now, we want to show that assumptions(L1)–(L5) and convergence criterion which was prepared in Theorem 7 are satisfied for integral equation (4).Noting thatγ2(x)=(1/2)x, B(x)=D(x)=x2, ∀x∈I, then (L1)–(L5) is true. For details see [6].In addition, we haveβ(x)=1, α1(x)=0, α2(x)=(1/50)x2, α3(x)=α4(x)=(1/100)x17, K(x,r)=8xr, h(x,r)=2x2r, and η1=η2=1/4; then N1=1, N2=1/50, N3=8, and N4=2, and then
(37)N2=150≤1-(η1+η2bN1)b(2+N3+N4)=124.
Therefore, the iterative method (31) converges to the exact solution of this equation. Now, based on Sinc quadrature scheme we can have some successive approximations for solution of this example for κ=1/2. Absolute error for nth approximation and N points quadrature method is defined by
(38)en,M=max⁡x∈[0,1]{|yn+1,M(x)-yn,M(x)|};n=1,2,3,….
For different values of n absolute errors are depicted in Figure 1 and approximate solution for n=30 is depicted in Figure 2.Figure 1
Absolute errors of Example1. For different values of iteration n.Figure 2
Approximate solution for Example1.Example 2.
In Example 1 of [19], authors considered the integrodifferential equation
(39)dydx=1-13x3+∫01‍x3y2(z)dz,y(0)=0.
The exact solution is y(x)=x. Maximum absolute error for each iteration and different values of quadrature points are depicted in Figures 3 and 4. By comparing these results with the numerical results given in [19] in Table 2, efficiency and accuracy of current approach are guaranteed.Figure 3
Maximum absolute error related to each iteration in Example2.Figure 4
Maximum error related to different quadrature points in Example2.


## 6. Conclusion

In this paper, we apply a numerical approach by Sinc function for reaching the estimated solution of integrodifferential equation with enteral boundary condition and with delay parameter. To reach this aim we change this problem to a functional enteral equation. The Sinc estimation has exponential convergence rate such asO(-ceM1/2) that this property is an advantage. so we applied it to solve our problem by using collocation method. Finally, some examples are solved by this numerical method to show the efficiency and accuracy of Sinc estimation. It is worthy to note that this method can be used for solving integrodifferential equations with integral boundary conditions with deviating arguments arising in all sciences such as chemistry, physics, and other fields of applied mathematics.



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*Source: 564632-2014-05-08.xml*