NUMERICAL SOLUTION OF NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS VIA DIRECT METHOD USING TRIANGULAR FUNCTIONS

 Dr.K.R.Salini[1] and Dr.S.Bamini[2]

[1] Guest Lecturer, PG & Department of Mathematics, Government Arts & Science College, Kariyampatti, Tirupattur, Tamilnadu, India.

Email: shalinimanoj1985@gmail.com

[2] Research Coordinator, Assistant Professor, PG & Research Department of Mathematics, Marudhar Kesari Jain College For Women, Vaniyambadi, Tamilnadu, India.

Email: saransham07@gmail.com

Abstract

We determine the numerical solution of the specific nonlinear Voltera-Fredholm Integro-Differential Equation is proposed. The method is based on new vector forms for representation of triangular functions and its operational matrix.

Introduction

A set of basic functions and an appropriate projection method or a direct method. These methods often transform an integro-differential equation to a linear or nonlinear system of algebraic equations which can be solved by direct or iterative methods.

Consider a Volterra-Fredhlom Integro-differential equations of the form 𝑥′(𝑠)+𝑞(𝑠)𝑥(𝑠)+𝜆1∫𝑘1(𝑠,𝑡)𝐹(𝑥(𝑡))𝑑𝑡10+𝜆2∫𝑘2(𝑠,𝑡)𝐺(𝑥(𝑡))𝑑𝑡10=𝑦(𝑠)

𝑥(0)=𝑥0

Where the functions 𝐹(𝑥(𝑡)) and 𝐺(𝑥(𝑡)) are polynomials of 𝑥(𝑡) with constant coefficients.

For convenience, we put 𝐹(𝑥(𝑡))=[𝑥(𝑡)]𝑛1 and 𝐺(𝑥(𝑡))=[𝑥(𝑡)]𝑛2, where 𝑛1 & 𝑛2 are positive integers.

For 𝑛1,𝑛2=1, equation (1.1) is a linear integro- differential equation.

We present new vector forms of triangular functions (TFs), operational matrix of integration, expansion of functions of one and two variables with respect to TFs and other TFs properties.

By using new representations, a nonlinear integro-differential equation can be easily reduced to a nonlinear system of algebraic equations.

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Triangular Functions

Definition

Two 𝑚-sets of triangular functions are defined over the intercal [0,𝑡) as 𝑇1𝑖(𝑡)={1−𝑡−𝑖ℎℎ𝑖ℎ≤𝑡<(𝑖+1)ℎ0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑇2𝑖(𝑡)={𝑡−𝑖ℎℎ𝑖ℎ≤𝑡<(𝑖+1)ℎ0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Where 𝑖=0,1,2…𝑚−1 with a positive integer value for m.

Also consider ℎ=𝑇/𝑚, and 𝑇1𝑖 as the 𝑖𝑡ℎ left-handed triangular function and 𝑇2𝑖 as

the 𝑖𝑡ℎ right-handed triangular function.

It is assumed that 𝑇=1, so TFs are defined over [0,1) and ℎ=1/𝑚.

From the definition of TFs, it is clear that triangular functions are disjoint, orthogonal

and complete.

We can write ∫𝑇1𝑖10(𝑡)𝑇1𝑗(𝑡)𝑑𝑡=∫𝑇2𝑖10(𝑡)𝑇2𝑗(𝑡)𝑑𝑡={ℎ3𝑖=𝑗0𝑖≠𝑗

Also, ∅𝑖(𝑡)=𝑇1𝑖(𝑡)+𝑇2𝑖,𝑖=0,1,…𝑚−1

Where ∅𝑖(𝑡) is the 𝑖th block-pulse function defined as ∅𝑖(𝑡)={1𝑖ℎ≤𝑡<(𝑖+1)ℎ0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Where 𝑖=0,1,…𝑚−1.

Vector Forms

Consider the first 𝑚 terms of the left-handed triangular functions and the first 𝑚 terms of the right- handed triangular functions and write them concisely as 𝑚-vectors. 𝑇1(𝑡)=[𝑇10(𝑡),𝑇11(𝑡),…𝑇1𝑚−1(𝑡)]𝑇

𝑇2(𝑡)=[𝑇20(𝑡),𝑇21(𝑡),…𝑇2𝑚−1(𝑡)]𝑇

Where 𝑇1(𝑡) and 𝑇2(𝑡) are called left-handed triangular functions (LHTF) and right-handed triangular functions (RHTF) vector, respectively.

The product of two TFs vectors are

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𝑇1(𝑡)𝑇1𝑇(𝑡)≅( 𝑇10(𝑡)0⋯⋯⋯00𝑇11(𝑡)⋯⋯⋯0⋮⋮0⋮⋮0⋮⋮⋯⋮⋮⋯⋮⋮⋯⋮⋮𝑇1𝑚−1(𝑡))

𝑇2(𝑡)𝑇2𝑇(𝑡)≅( 𝑇20(𝑡)0⋯⋯⋯00𝑇21(𝑡)⋯⋯⋯0⋮⋮0⋮⋮0⋮⋮⋯⋮⋮⋯⋮⋮⋯⋮⋮𝑇2𝑚−1(𝑡)) 𝑇1(𝑡)𝑇2𝑇(𝑡)=0

𝑇2(𝑡)𝑇1𝑇(𝑡)=0

Where 0 is the 𝑧𝑒𝑟𝑜 𝑚×𝑚 matrix.

Also ∫𝑇110(𝑡)𝑇1𝑇(𝑡)𝑑𝑡=∫𝑇2(𝑡)𝑇1𝑇(𝑡)𝑑𝑡10≅ℎ3𝐼

∫𝑇110(𝑡)𝑇2𝑇(𝑡)𝑑𝑡=∫𝑇2(𝑡)𝑇1𝑇(𝑡)𝑑𝑡10≅ℎ6𝐼

In which 𝐼 is an 𝑚×𝑚 identity matrix.

TFs Expansion

The expansion of a function 𝑓(𝑡) over [0,1) with respect to TFs written as 𝑓(𝑡)≅Σ𝑐𝑖𝑚−1𝑖=0𝑇1𝑖(𝑡)+Σ𝑑𝑖𝑚−1𝑖=0𝑇2𝑖(𝑡) 𝑓(𝑡)=𝑐𝑇𝑇1(𝑡)+𝑑𝑇𝑇2(𝑡)

Where we may put 𝑐𝑖=𝑓(𝑖ℎ) and 𝑑𝑖=𝑓((𝑖+𝑖)ℎ) for 𝑖=0,1,2,..𝑚−1.

Operational Matrix of Integration

Expressing ∫𝑇1(𝜏)𝑑𝜏≅𝑃1𝑇1(𝑠)+𝑃2𝑇2(𝑠)𝑠0

∫𝑇2(𝜏)𝑑𝜏≅𝑃1𝑇1(𝑠)+𝑃2𝑇2(𝑠)𝑠0

Where 𝑃1𝑚×𝑚 and 𝑃2𝑚×𝑚 are called operational matrices of integration in TFs domain and represented as follows 𝑃1=ℎ2( 011⋯1001⋯10⋮00⋮00⋮0⋯⋱⋯1⋮0)

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𝑃2=ℎ2( 111⋯1011⋯10⋮00⋮00⋮0⋯⋱⋯1⋮1)

So the integral of any function 𝑓(𝑡) can be approximated as ∫𝑓(𝜏)𝑑𝜏≅𝑐𝑇𝑇1(𝑠)+𝑑𝑇𝑇2(𝑠)𝑑𝜏𝑠0 ∫𝑓(𝜏)𝑑𝜏≅(𝑐+𝑑)𝑇𝑃1𝑇1(𝑠)+(𝑐+𝑑)𝑇𝑃2𝑇2(𝑠)𝑠0

New Representation of TFs Vector Forms and Other Properties

We define a new representation of TFs vector forms. Then, some characteristics of TFs are presented using the new definition.

Definition and Expansion

Let 𝑇(𝑡) be a 2𝑚-vector defined as 𝑇(𝑡)=(𝑇1(𝑡)𝑇2(𝑡)),0<𝑡<1

Where 𝑇1(𝑡) and 𝑇2(𝑡) have been defined in (1.2).

Now, the expansion of 𝑓(𝑡) with respect to TFs can be written as 𝑓(𝑡)≅𝐹1𝑇𝑇1(𝑡)+𝐹2𝑇𝑇2(𝑡)

≅𝐹𝑇𝑇(𝑡)

Where 𝐹1 &𝐹2 are TFs co-efficients with

𝐹1𝑖=𝑓(𝑖ℎ)& 𝐹2𝑖=𝑓((𝑖+1)ℎ),

For 𝑖=0,1,…𝑚−1.

Also, 2𝑚-vector 𝐹 is defined as 𝐹=(𝐹1𝐹2)

Now, assume that 𝑘(𝑠,𝑡) is a function of two variables. It can be expanded with respect to TFs as follows

𝑘(𝑠,𝑡)≅𝑇𝑇(𝑠)𝐾𝑇(𝑡)

Where 𝑇(𝑠)& 𝑇(𝑡) are 2𝑚1&2𝑚2 dimensional triangular functions and 𝐾 is a

2𝑚1×2𝑚2 TFs coefficient matrix.

For convenience, we put 𝑚1=𝑚2=𝑚. So matrix 𝐾 can be written as

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𝐾=((𝐾11)𝑚×𝑚(𝐾12)𝑚×𝑚(𝐾21)𝑚×𝑚(𝐾22)𝑚×𝑚)

Where 𝐾11,𝐾12,𝐾21,𝐾22 can be compted by sampling the function 𝑘(𝑠,𝑡) at points 𝑠𝑖 & 𝑡𝑗 such that 𝑠𝑖=𝑖ℎ & 𝑡𝑗=𝑗ℎ, for 𝑖,𝑗=0,1,…𝑚. (𝐾11)𝑖𝑗=𝑘(𝑠𝑖,𝑡𝑗),𝑖=0,1,…𝑚−1,𝑗=0,1,…𝑚−1 (𝐾12)𝑖𝑗=𝑘(𝑠𝑖,𝑡𝑗),𝑖=0,1,…𝑚−1,𝑗=0,1,…𝑚 (𝐾21)𝑖𝑗=𝑘(𝑠𝑖,𝑡𝑗),𝑖=0,1,…𝑚,𝑗=0,1,…𝑚−1 (𝐾22)𝑖𝑗=𝑘(𝑠𝑖,𝑡𝑗),𝑖=0,1,…𝑚,𝑗=0,1,…𝑚

Product Properties

Let 𝑋 be an 2𝑚-vertor which can be written as 𝑋𝑇=(𝑋1𝑇 𝑋2𝑇) such that 𝑋1 & 𝑋2 are 𝑚-vectors.

It can be concluded from (1.3) & (1.4) that 𝑇(𝑡)𝑇𝑇(𝑡)𝑋=(𝑇1(𝑡)𝑇2(𝑡))(𝑇𝑇(𝑡)𝑇2𝑇(𝑡))(𝑋1𝑋2)

≅(𝑑𝑖𝑔(𝑇1(𝑡))𝑜𝑚×𝑚𝑜𝑚×𝑚𝑑𝑖𝑔(𝑇2(𝑡)))(𝑋1𝑋2)

=𝑑𝑖𝑔 (𝑇(𝑡))𝑋

=𝑑𝑖𝑔(𝑋)𝑇(𝑡)

Therefore,

𝑇(𝑡)𝑇𝑇(𝑡)𝑋≅𝑋̅ 𝑇(𝑡)

Where 𝑋̅=𝑑𝑖𝑔(𝑋) is an 2𝑚×2𝑚 diagonal matrix.

Let 𝐵 be a 2𝑚×2𝑚 matrix as 𝐵=((𝐵11)𝑚×𝑚(𝐵12)𝑚×𝑚(𝐵21)𝑚×𝑚(𝐵22)𝑚×𝑚)

So, it can be similarly concluded from equations (1.3) and (1.4) that 𝑇(𝑡)𝐵𝑇(𝑡)=𝑇1𝑇(𝑡)𝑇2𝑇(𝑡)(𝐵11𝐵12𝐵21𝐵22)(𝑇1(𝑡)𝑇2(𝑡)) ≅𝑇1𝑇(𝑡)𝐵11𝑇1(𝑡)+𝑇2𝑇(𝑡)𝐵22𝑇2(𝑡) ≅𝐵̂11𝑇𝑇1(𝑡)+𝐵̂22𝑇𝑇2(𝑡)

Where 𝐵̂11 and 𝐵̂22 are 𝑚−vectors with elements equal to the diagonal entries of matrices 𝐵11 and 𝐵22 respectively.

Therefore,

𝑇𝑇(𝑡)𝐵𝑇(𝑡)≅𝐵̂ 𝑇𝑇(𝑡)

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In which 𝐵̂ is a 2𝑚- vector with elements equal to the diagonal entries of matrix 𝐵.

It is immediately concluded from equation (1.5) that ∫𝑇10(𝑡)𝑇𝑇(𝑡)𝑑𝑡=∫(𝑇1(𝑡)𝑇2(𝑡))𝑇1𝑇(𝑡)𝑇2𝑇(𝑡)𝑑𝑡10

=∫(𝑇1(𝑡)𝑇1𝑇(𝑡)𝑇1(𝑡)𝑇2𝑇(𝑡)𝑇2(𝑡)𝑇1𝑇(𝑡)𝑇2(𝑡)𝑇2𝑇(𝑡))𝑑𝑡10

≅(ℎ3𝐼𝑚×𝑚ℎ6𝐼𝑚×𝑚ℎ6𝐼𝑚×𝑚ℎ3𝐼𝑚×𝑚)

Therefore,

∫𝑇10(𝑡)𝑇𝑇(𝑡)𝑑𝑡≅𝐷

Where 𝐷 is the following 2𝑚×2𝑚 matrix. 𝐷=ℎ3( 10⋯012⁄0⋯001⋯0012⁄⋯0⋮012⁄0⋮0⋮0012⁄⋮0⋱⋯⋯⋯⋱⋯⋮100⋮12⁄⋮010⋮0⋮001⋮0⋱…⋯⋯⋱⋯⋮12⁄00⋮1)

Operational Matrix

Expressing ∫𝑇(𝜏)𝑑𝜏𝑠0 in terms of 𝑇(𝑠), we can write ∫𝑇(𝜏)𝑑𝜏𝑠0=∫(𝑇1(𝜏)𝑇2(𝜏))𝑑𝜏𝑠0

≅(𝑃1𝑇1(𝑠)+𝑃2𝑇2(𝑠)𝑃1𝑇1(𝑠)+𝑃2𝑇2(𝑠))

=(𝑃1𝑃2𝑃1𝑃2)(𝑇1(𝑠)𝑇2(𝑠))

So, ∫𝑇(𝜏)𝑑𝜏𝑠0≅𝑃𝑇(𝑠)

Where 𝑃2𝑚×2𝑚, operational matrix of 𝑇(𝑠) is

𝑃=(𝑃1𝑃2𝑃1𝑃2) (

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Where 𝑃1 & 𝑃2 are given by

The integral of any function 𝑓(𝑡) can be approximated as ∫𝑓(𝜏)𝑑𝜏𝑠0≅∫𝐹𝑇𝑇(𝜏)𝑑𝜏𝑠0

≅𝐹𝑇𝑃𝑇(𝑠)

Solving Nonlinear Integro-Differential Equation

Consider the following nonlinear Volterra-Fredhlom integro-differential equation

{𝑥′(𝑠)+𝑞(𝑠)𝑥(𝑠)+𝜆1∫𝑘1(𝑠,𝑡)[𝑥(𝑡)]𝑛1𝑑𝑡+𝑠0𝜆2∫𝑘2(𝑠,𝑡)[𝑥(𝑡)]𝑛2𝑑𝑡=𝑦(𝑠)𝑠0𝑥(0)=𝑥0, 0≤𝑠<1,𝑛1,𝑛2≥1

Where the parameters 𝜆1 and 𝜆2 and ℒ2 functions 𝑞(𝑠),𝑦(𝑠),𝑘1(𝑠,𝑡),𝑘2(𝑠,𝑡) are known but 𝑥(𝑠) is not.

The appearance of initial condition equation. This is necessary to ensure the existence of a solution.

Approximating functions

𝑥(𝑠),𝑥′(𝑠),𝑞(𝑠),𝑦(𝑠),[𝑥(𝑡)]𝑛1,[𝑥(𝑡)]𝑛2,𝑘1(𝑠,𝑡),𝑘2(𝑠,𝑡)

with respect to TFs, 𝑥(𝑠)≅𝑥𝑇(𝑠)𝑋 𝑥′(𝑠)≅𝑋′𝑇𝑇(𝑠)=𝑇𝑇𝑋′

𝑞(𝑠)≅ 𝑄𝑇𝑇(𝑠)=𝑇𝑇(𝑠)𝑄

𝑦(𝑠)≅ 𝑌𝑇𝑇(𝑠)=𝑇𝑇(𝑠)𝑌 [𝑥(𝑡)]𝑛1≅𝑋𝑛1𝑇𝑇(𝑠)=𝑇𝑇(𝑠)𝑋𝑛1 [𝑥(𝑡)]𝑛2≅𝑋𝑛2𝑇𝑇(𝑠)=𝑇𝑇(𝑠)𝑋𝑛2 𝑘1(𝑠,𝑡)≅𝑇𝑇𝐾1𝑇(𝑡)

𝑘2(𝑠,𝑡)≅𝑇𝑇𝐾2𝑇(𝑡)

Where 2𝑚-vectors 𝑋,𝑋′,𝑄,𝑌,𝑋𝑛1,𝑋𝑛2 and 2𝑚×2𝑚 matrices 𝐾1 & 𝐾2 are TFs coefficients of

𝑥(𝑠),𝑥′(𝑠),𝑞(𝑠),𝑦(𝑠),[𝑥(𝑡)]𝑛1,[𝑥(𝑡)]𝑛2,𝑘1(𝑠,𝑡),𝑘2(𝑠,𝑡) respectively.

Lemma

Let 2𝑚-vectors 𝑋 and 𝑋𝑛 be TFs coefficients of 𝑥(𝑠) and [𝑥(𝑠)]𝑛 respectively. If

𝑋=(𝑋1𝑇𝑋2𝑇)𝑇=(𝑋10,𝑋11,…,𝑋1𝑚−1,𝑋20,𝑋21…𝑋2𝑚−1)𝑇, then

𝑋𝑛=(𝑋10,𝑋11,…,𝑋1𝑚−1,𝑋20,𝑋21…𝑋2𝑚−1)𝑇

Where 𝑛≥1 is a positive integer.

Proof:

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When 𝑛=1, , follows at once from [𝑥(𝑠)]𝑛=𝑥(𝑠).

Suppose that (4.16), holds for 𝑛, we shall deduce it for 𝑛+1.

Since [𝑥(𝑠)]𝑛+1≅(𝑋𝑇𝑇(𝑠)).(𝑋1𝑇𝑇(𝑠))

=𝑋𝑇𝑇(𝑠)𝑇𝑇(𝑠)𝑋𝑛 =𝑋𝑇𝑋𝑛̃𝑇(𝑠)

we obtain 𝑋𝑇𝑋𝑛̃=(𝑋10𝑛+1,𝑋11𝑛+1,……,𝑋1𝑚−1𝑛+1,𝑋20𝑛+1,𝑋21𝑛+1,……,𝑋2𝑚−1𝑛+1)𝑇

Equation holds for 𝑛+1.

Components of 𝑿𝒏 can be Computed in terms of Components of Unknown Vector 𝑿

We substitute (1.15) into (1.14), we have 𝑌𝑇𝑇(𝑠)≅𝑋′𝑇𝑇(𝑠)+𝑄𝑇𝑇(𝑠)𝑇𝑇(𝑠)𝑋+𝜆1𝑇𝑇(𝑠)𝐾1∫𝑇(𝑡)𝑇𝑇(𝑡)𝑋𝑛1𝑑𝑡+𝑠0𝜆2𝑇𝑇(𝑠)𝐾2∫𝑇(𝑡)𝑇𝑇(𝑡)𝑋𝑛2𝑑𝑡𝑠0

it follows that 𝑌𝑇𝑇(𝑠)≅𝑋′𝑇𝑇(𝑠)+(𝑄̃ 𝑇(𝑠))𝑇𝑋+𝜆1𝑇𝑇(𝑠)𝐾1𝑋̃𝑛1∫𝑇(𝑡)𝑑𝑡+𝜆2𝑇𝑇(𝑠)𝐾2𝐷𝑋𝑛2𝑠0

Using Operational matrix 𝑃,

𝑌𝑇𝑇(𝑠)≅𝑋′𝑇𝑇(𝑠)+𝑋𝑇𝑄̃ 𝑇(𝑠)+𝜆1𝑇𝑇(𝑠)𝐾1𝑋̃𝑛1𝑃𝑇(𝑠)+𝜆2(𝐾2𝐷𝑋𝑛2)𝑇𝑇(𝑠)

In which 𝜆1𝐾1𝑋̃𝑛1𝑃 is a 2𝑚×2𝑚 matrix

𝑇𝑇(𝑠)𝜆1𝐾1𝑋̃𝑛1𝑃𝑇(𝑠)≅𝑋𝑛1𝑇̂𝑇(𝑠)

Where 𝑋̃𝑛1 is an 2𝑚-vector with components equal to the diagonal entries of the matrix 𝜆1𝐾1𝑋̃𝑛1𝑃.

𝑌𝑇𝑇(𝑠)≅𝑋′𝑇𝑇(𝑠)+𝑋𝑇𝑄̃ 𝑇(𝑠)+𝑋𝑛1𝑇̂𝑇(𝑠)+𝜆2(𝐾2𝐷𝑋𝑛2)𝑇𝑇(+𝑄̃ 𝑋+𝑋𝑛1̂+ 𝜆2𝐾2𝐷𝑋𝑛2≅𝑌

Where 𝑄̃ is a diagonal matrix, so 𝑄̃ 𝑇=𝑄̃.

𝑋′ must be computed in terms of 𝑋. 𝑥(𝑠)−𝑥(0)=∫𝑥′(𝜏)𝑑𝜏𝑠0

≅∫𝑋′𝑇𝑇(𝜏)𝑑𝜏𝑠0 ≅𝑋′𝑇𝑃𝑇(𝑠) 𝑥(𝑠)≅𝑋′𝑇𝑃𝑇(𝑠)+𝑋0𝑇𝑇(𝑠)

Where 𝑋0 is the 2𝑚- vector of the form

Journal of the Maharaja Sayajirao University of Baroda ISSN: 0025-0422

75

Volume-56, No.1 (IV) 2022

𝑋0=[𝑥0,𝑥0,…𝑥0]𝑇

Consequently,

𝑋≅𝑃𝑇𝑋′+𝑋0

Now, combining (1.19) and (1.20) and replacing ≅ with =, it follows that

(𝐼+𝑃𝑇𝑄̃)𝑋+𝑃𝑇𝑋𝑛1̂+𝜆2𝑃𝑇𝐾2𝐷𝑋𝑛2≅𝑃𝑇𝑌+𝑋0

Equation is a nonlinear system of 2𝑚 algebraic equations for the 2𝑚 unknowns 𝑋10,𝑋11,…𝑋1𝑚−1,𝑋20,𝑋21,…𝑋2𝑚−1.

Components of 𝑋𝑇=(𝑋1𝑇 𝑋2𝑇) can be obtainbed by an iterative method.

Hence, an approximate solution 𝑥(𝑠)≅𝑋𝑇𝑇(𝑠)

𝑥(𝑠)≅𝑋1𝑇𝑇1(𝑠)+𝑋2𝑇𝑇2(𝑠)

Can be computed for equation (1.14) without using any projection method.

Conclusion

We investigated the numerical solution of nonlinear and singularly perturbed for Vulture integro-differential equations. Also, numerical solution of high-order and fractional nonlinear Volterra-Fredholm integro-differential equations and its error analysis are discussed. Finally, some numerical results of integro-differential equations are presented to illustrated the efficiency and accuracy of the proposed methods.

Reference

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