Available via license: CC BY 4.0

Content may be subject to copyright.

Citation: González-Rodelas, P.;

Pasadas, M.; Kouibia, A.; Mustafa, B.

Numerical Solution of Linear

Volterra Integral Equation Systems of

Second Kind by Radial Basis

Functions. Mathematics 2022,10, 223.

https://doi.org/10.3390/

math10020223

Academic Editors: Sara Remogna,

Domingo Barrera, María José Ibáñez

and Simeon Reich

Received: 30 November 2021

Accepted: 4 January 2022

Published: 12 January 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional afﬁl-

iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

mathematics

Article

Numerical Solution of Linear Volterra Integral Equation

Systems of Second Kind by Radial Basis Functions

Pedro González-Rodelas †, Miguel Pasadas *,† , Abdelouahed Kouibia †and Basim Mustafa †

Department of Applied Mathematics, Granada University, 18071 Granada, Spain; prodelas@ugr.es (P.G.-R.);

kouibia@ugr.es (A.K.); bmustafa@correo.ugr.es (B.M.)

*Correspondence: mpasadas@ugr.es

† These authors contributed equally to this work.

Abstract:

In this paper we propose an approximation method for solving second kind Volterra

integral equation systems by radial basis functions. It is based on the minimization of a suitable

functional in a discrete space generated by compactly supported radial basis functions of Wendland

type. We prove two convergence results, and we highlight this because most recent published papers

in the literature do not include any. We present some numerical examples in order to show and justify

the validity of the proposed method. Our proposed technique gives an acceptable accuracy with

small use of the data, resulting also in a low computational cost.

Keywords: Volterra integral equations system; radial basis functions; variational methods

1. Introduction

A considerable large amount of research literature and books on the theory and applica-

tions of Volterra’s integral equations have emerged over many decades since the apparition

of Volterra’s book “Leçons sur les équations intégrales et intégro-différentielles” [

1

] in 1913.

The applications include elasticity, plasticity, semi-conductors, scattering theory, seis-

mology, heat and mass conduction or transfer, metallurgy, ﬂuid ﬂow dynamics, chemical

reactions, population dynamics, and oscillation theory, among many others (see for exam-

ple [

2

]). Other important references more related with the numerics of this type of equation

are [3,4].

In fact, Volterra integral equations (VIEs) appear naturally when we try to transform

an initial value problem into integral form, so that the solution of this integral equation

is usually much easier to obtain than the original initial value problem. In the same way,

some nonlinear Volterra integral equations are equivalent to an initial-value problem for a

system of ordinary differential equations (ODEs). So, some authors (like for example [

5

])

have sought to exploit this connection for the numerical solution of the integral equations

as well, since very effective ODE codes are widely available.

Volterra integral equations arise in many usual applications of technology, engineering

and science in general: as in population dynamics, the spread of epidemics, some Dirichlet

problems in potential theory, electrostatics, mathematical modeling of radioactive equilib-

rium, the particle transport problems of astrophysics and reactor theory, radiative energy

and/or heat transfer problems, other general heat transfer problems, oscillation of strings

and membranes, the problem of momentum representation in quantum mechanics, etc.

However, many other complex problems of mathematics, chemistry, biology, astrophysics

and mechanics, can be expressed in the terms of Volterra integral equations. Moreover,

some practical problems, where impulses arise naturally (like in population dynamics or

many biological applications) or are caused by some control system (like electric circuit

problems and simulations of semiconductor devices) can be modeled by a differential equa-

tion, an integral equation, an integro-differential equation, or a system of these equations

all combined.

Mathematics 2022,10, 223. https://doi.org/10.3390/math10020223 https://www.mdpi.com/journal/mathematics

Mathematics 2022,10, 223 2 of 15

The systems of integral and/or integro-differential equations are usually difﬁcult to

solve analytically, in particular systems of Volterra integral non-linear equations or with

variable coefﬁcients; so a numerical method is often needed. In such cases, it is required to

approximate the solutions; and many different numerical techniques have been developed

and presented during decades of research, with appropriate combinations of numerical

integration and interpolation procedures (see the references [

3

,

6

], among others). In order

to approximate numerically the solution of general integral equations, the predominant

technique have been the use of some kind of piecewise constant basis functions (PCBFs) (see

for example [

7

], among many others); Chebyshev polynomial ([

8

] and others). However,

after a long period of time many other techniques have attracted much attention recently;

like wavelets theory, started with the introduction of Haar function in 1910, and from

1990’s (see [

9

]) also many wavelet type methods have been applied for solving integral

equations. Haar wavelets, despite its relative simplicity, have many valued properties: as

its compact support and orthogonality properties. So they can be used for the solution of

differential and integro-differential equations related with signal and image processing,

for example. They have been also used to solve linear and nonlinear integral equations by

Aziz et. al. [10]

, Babolianet. al. [

11

], Lepik [

12

], Maleknejad et al. [

13

], Farshid Mirzaae [

14

],

among others. More recently, several numerical methods based on different triangular type

and delta orthogonal functions were designed for approximating the solution of integral

and/or integro-differential Volterra equations (see for example [

15

–

17

], and the references

therein). All these publications have demonstrated and revealed that these techniques

based on PCBF and wavelets are effective to obtain the solution of such integral equations.

Particularly, systems of linear integral equations, and their exact or approximate

solutions, are of great importance in science and engineering. There are several numerical

methods for solving systems of linear Volterra integral equations of the second kind, and

they have been often solved by classical numerical and analytical methods: such as Galerkin

and Finite Element methods, collocation and spectral methods, Taylor or Power series

and expansion methods, transforming the equations into a linear or nonlinear system of

algebraic equations, and so on. However, new methods also have been applied to solve

them, like the homotopy perturbation method [

18

], Adomian decomposition method (and

many others) [

19

], use of Legendre wavelets [

20

] or hybrid Legendre and block pulse

functions [

21

], Chebyshev polynomials [

22

,

23

], etc. Berenguer et al. [

24

,

25

] have solved

them with the aid of a combination of analytical methods and bi-orthogonal systems in

Banach spaces, Sahn et al. [

26

] have used Bessel polynomials method,

Malnekad et al. [15]

have employed delta basis functions (DBFs), Balakumar et al. [

27

] have applied the block-

pulse functions method, Li-Hong et al. [

28

] have applied reproducing kernel method.

Furthermore, there are also expansion methods for integral equations such as El-gendi’s

and Wolfe’s methods (see for example [

29

]). Additionally, the approximate solutions of

systems of integral equations that usually appear in problems of physics, biology and

engineering are based on numerical integration methods: such as Euler–Chebyshev or

Runge–Kutta methods (see for example [30]).

Concerning many other possible techniques to solve these types of integral equations,

Draidi and Qatanani [

31

] implemented a product Nystrom and sinc-collocation meth-

ods to solve Volterra integral equations with Carleman kernel; also Issa, Qatanani and

Daraghmeh [

32

] used a Taylor expansion and the variational iteration methods to give an

approximate solution of Volterra integral equations of the second kind.

Aggarwal et al. [33]

and Chauhan [

34

] used different integral transformations for obtaining the solutions of VIEs

of second kind. Mahgoub [

35

] solved constant coefﬁcient linear differential equations by

deﬁning the called Sawi transformation, but many other authors exploited this idea, or other

appropriate transforms, to deal with these types of integral or integro-differential equations.

Next, we are going to cite the most recent references from the last 3 or 4 years. In [

36

]

the authors present an approximation solution of system of Volterra integral equations of

second kind in an analytical way, using an Adomian decomposition method in Mathematica.

In [

37

] the authors propose a numerical algorithm based on Monte Carlo method for

Mathematics 2022,10, 223 3 of 15

approximating solutions of the system of Volterra integral equations. In [

38

] the authors

develop a numerical technique for the solution of 2D Volterra integral equations based on a

discretization method by using two-dimensional Bernstein’s approximations. In [

39

] the

authors discussed the solution of linear Volterra integral equations of second kind using

Mohand transform. In [

40

] the authors propose Bernstein polynomials to present effective

solution for the second kind linear Volterra integral equations with delay. In [

41

] the author

presents a method to solve numerically Volterra integral equations of the ﬁrst kind with

separable kernels.

In this work, we will present some speciﬁc variational methods adopted to study

and approximate systems of linear Volterra integral equations with the aid of Radial Basis

Functions (RBFs) of Wendland type. Wendland functions are compactly supported radial

basis functions, which makes calculations with them quite simple. However, the general

Wendland family of functions are deﬁned recursively, and to determine the actual functions

to use in any software implementation many calculations had to be done by hand or with

the aid of some symbolic software (see for example [

42

]). There are for the moment just

a few articles dealing with this type of techniques, like for example [

43

–

47

]; so we think

there is still a lot to investigate in this regard.

Our goal in this work is to devise an appropriate approach procedure that is capable

of solving this type of problem in a precise and efﬁcient way. We consider then the linear

Volterra equations system of the second kind as follows (see for example [1]):

x(t) = f(t) + Zt

0k(t,s)x(s)ds, 0 ≤s≤t≤1, (1)

where

x(t) = (x1(t), . . . , xn(t))>,

f(t) = ( f1(t), . . . , fn(t))>,

k(t,s) = (kij(t,s))1≤i,j≤n.

We assume that (1) has a unique continuous solution for appropriate functions

f

. In

any case, the equations system (1) can be re-written in operator form as an equation of

second kind

f= (I−K)x,

where

K

is an integral operator and

I

denotes the identity operator. It is usual to impose

certain assumptions on compactness on the operator

K

(see [

48

], Section 2.8.1) in order to

establish the existence and uniqueness of the solution of (1), that we will assume throughout

the work.

Moreover, in [

49

] the authors proposed another method to solve second kind Fredholm

integral equation systems, but the discrete functional space chosen in that article has been

the space of spline functions. While at ﬁrst glance it might seem that both works are

similar, especially in the way they are presented, the two methods are totally different,

not only be the fact that the discretization spaces are different (so we have adapted the

notations accordingly), while the proofs (except the very preliminary ones, that can be also

adapted), above all the proofs of the convergence results, are completely different, due to

their greater complexity.

The outline of the paper is as follows. In Section 2we brieﬂy recall some notations

and preliminaries. Section 3is devoted to establish the discretization space as a radial

basis functions space. The formulation of the minimization problem is realized in

Section 4

and two equivalent variational problems are given. Section 5is devoted to prove two

convergence results. Section 6deals with the description of the computation algorithm of

the discrete problem solution. In Section 7we present some numerical experiments and

ﬁnally, in Section 8we establish the conclusions of the work.

Mathematics 2022,10, 223 4 of 15

2. Notations and Preliminaries

Let

R+

0={x∈R:x≥

0

}

; and for

n≥

1, we denote by

h·in

and

h ·

,

· in

the

Euclidean norm and the inner product in Rn.

On the other hand, for

m≥

1, we designate by

Hm((

0, 1

)

;

Rn)

the Sobolev space of

order

m

of (classes of) functions

u∈L2((

0, 1

)

;

Rn)

together with all

j

-th derivative functions

u(j)of order j≤m, in the sense of distributions. This space is equipped with

• the semi–inner products, for any u,v∈Hm((0, 1);Rn),

(u,v)j=Z1

0hu(j)(t),v(j)(t)indt, 0 ≤j≤m,

• the corresponding semi–norms |u|j= (u,u)1

2

j, for 0 ≤j≤m,

• the inner product ((u,v))m=

m

∑

0

(u,v)j,

• and the corresponding norm kukm= ((u,u)) 1

2

m.

For any 1

≤i≤n

, let

ki

be a given function of the Sobolev vectorial functions space

Hm((0, 1)×(0, 1);Rn)and consider the matrix valued function

k(t,s) = (ki(t,s))1≤i≤n∈Hm((0, 1)×(0, 1);Rn,n),

together with the associated integral operator

Ku(t) = Zt

0hki(t,s),u(s)inds1≤i≤n

,t∈(0, 1),∀u∈Hm((0, 1);Rn).

Let

Rn,p

be the space of real matrices of

n

lines and

p

columns, equipped with the

inner product

hA,Bin,p=

n

∑

i=1

p

∑

j=1

aij bij,∀A= (aij )1≤i≤n

1≤j≤p,B= (bij )1≤i≤n

1≤j≤p∈Rn,p,

and the corresponding norm hAin,p=hA,Ai1

2

n,p.

3. Discretization Space

For the remainder of the work, we are going to consider a space of ﬁnite dimension,

where we will formulate and solve a discrete approximation problem. The discrete func-

tional space we have chosen is the radial basis functions space with compact support,

namely the radial basis function space generated by the Wendland functions (see [50]).

Deﬁnition 1.

Given a continuous function

φ:R+

0→R

, a subset

Ω⊂Rd

,

d≥

1, and a point

ξ∈Ω

, the radial function deﬁned on

Ω

from the function

φ

with center

ξ

is the continuous function

Φξ:Ω→R given by

Φξ(x) = φ(hx−ξid).

Then Φξonly depends of the distance to ξ.

Deﬁnition 2. Given a centers set Ξ={ξ1, . . . , ξN}the linear space generated by the functions

{φ(h· − ξ1i)d, . . . , {φ(h· − ξNi)d}

is called a radial basis functions space.

Mathematics 2022,10, 223 5 of 15

Deﬁnition 3.

For a function

u∈C([

0, 1

]

;

Rn)

, the radial basis function interpolating

u

on a set

of distinct centers TN={t1, . . . , tN} ⊂ [0, 1]is given by

su,TN(t) =

N

∑

i=1

αiφ(|t−ti|),t∈[0, 1],

where

φ:R+

0→R

is a continuous function and the coefﬁcients

α1

,

. . .

,

αN∈Rn

are determined

by the interpolation conditions

su,TN(ti) = u(ti), 1 ≤i≤N.

In [

50

] H. Wendland introduced a family of compactly supported radial basis functions

in the following way: let the operator Iand its inverse Dfor r≥0 be given by

(Iφ)(r) = Z∞

rtφ(t)dt,

(Dφ)(r) = −1

rφ0(r),

for any differentiable function φ:R+

0→R.

Given the truncated power function φ`(r) = (1−r)`

+, we deﬁne

φd,k=Ikφbd

2c+k+1,

where bxcdenotes the largest integer less than or equal to x.

Theorem 1.

([

50

], Theorem 1.2) The functions

φd,k

induce positive deﬁnite functions on

Rd

of

the form

φd,k(r) = pd,k(r), 0 ≤r≤1,

0, r>1,

with a univariate polynomial

pd,k

of degree

bd

2c+

3

k+

1. They possess continuous derivatives

up to order 2

k

, and they are of minimal degree for a given constant factor, uniquely determined by

this setting.

Thus, these functions are the natural candidates for interpolation by compactly sup-

ported radial basis functions, and they are called the Wendland’s functions.

For the remainder of the work we suppose 0

≤k≤N−

1, and we take

φ=φ1,k

in

Deﬁnition 3.

Table 1shows the Wendland functions φ1,kfor k=0, 1,2, and its continuity order.

Table 1. The Wendland functions φ1,kfor k=0, 1, 2 and its continuity order.

kWendland Function Continuity Order

k=0φ1,0(r) = (1−r)+C0

k=1φ1,1(r).

= (1−r)3

+(3r+1)C2

k=2φ1,2(r).

= (1−r)5

+(8r2+5r+1)C4

Let

h=sup

t∈[0,1]

min

1≤i≤N|t−ti|. (2)

From ([

50

], Theorem 2.1) we can afﬁrm that

φ1,k∈C2k([

0, 1

])

and the corresponding

native space is

Hk+1([

0, 1

])

. Finally, from ([

50

], Theorem 2.1) and ([

51

], Theorem 4.1) we

conclude that there exists C>0 such that

Mathematics 2022,10, 223 6 of 15

ku−su,TNkL∞((0,1);Rn)≤Ckukk+1hk+1

2,∀u∈Hk+1([0, 1];Rn),

and

|u−su,TN|j≤Chk+1−jkukk+1, 0 ≤j≤k+1, ∀u∈Hk+1([0, 1];Rn). (3)

Let

SN

be the space of the restrictions of functions on

[

0, 1

]

of the functional space

generated by the radial basis functions

{φ1,k(| · −t1|)

,

. . .

,

φ1,k(| · −tN|)}

and

SN= (SN)n

.

Then SN⊂Hk+1((0, 1);Rn)∩C2k([0, 1];Rn).

4. Formulation of the Problem

We can deﬁne the operator ρ:Hk+1((0, 1);Rn)→RN,ngiven by

ρv= ((I−K)v(ti))1≤i≤N.

Let assume that

f∈Hk+1((

0, 1

)

;

Rn)

and consider the afﬁne variety

HN={u∈SN:

ρu= ( f(ti))1≤i≤N}and the linear subspace H0

N={u∈SN:ρu=0∈RN,n}.

Proposition 1.

The set

HN

is a nonempty closed bounded convex subset of

SN

. Moreover it is an

afﬁne variety associated with the linear subspace H0

N.

Proof. By adapting the notations, as in the proof of Proposition 4.1 of [49].

Lemma 1. The application << ·,·>>:Hk+1((0, 1);Rn)×Hk+1((0, 1);Rn)→Rdeﬁned by

<< u,v>>=hρu,ρviN,n+ ((I−K)u,(I−K)v)k+1

is an inner product on

Hk+1((

0, 1

)

;

Rn)

and its associated norm, given by

[[u]] =<< u

,

u>> 1

2

, is

equivalent to the usual Sobolev norm k · kk+1.

Proof.

By adapting the notations as in the proof of Lemma 4.2 of [

49

] and using ([

48

],

Theorem 7.3.12) the proof can be obtained.

Deﬁnition 4.

We say that

uN∈HN

is an approximating radial basis function relative to

TN

,

ρ

and fif uNis a solution of the following minimization problem:

Find uN∈HNsuch that ∀v∈HN,J(uN)≤J(v), (4)

where J :Hk+1((0, 1);Rn)→Ris given by

J(v) = |(I−K)v|2

k+1.

Theorem 2.

Problem (4) has a unique solution

uN∈HN

which is the unique solution of the

variational problem

∀v∈H0

N,((I−K)uN,(I−K)v)k+1=0. (5)

Proof.

From Proposition 1and ([

48

], Theorem 3.4.3) we can deduce that there exists a

unique uN∈HN, which is the projection of 0on HNsuch that

[[uN]] ≤[[v]],∀v∈HN

and verifying

∀w∈HN,<< −uN,w−uN>>≤0,

that is

∀v∈H0

N,<< −uN,v>>≤0

Mathematics 2022,10, 223 7 of 15

and, taking into account that H0

Nis a vector space, we obtain that

∀v∈H0

N,<< uN,v>>=0.

Therefore (5) holds. Finally,

uN

is the unique solution of (4) since

J(v) = [[v]]2−

hρfi2

N,n, for any v∈HN.

Theorem 3. There exists a unique λ∈RN,nsuch that

∀v∈SN,((I−K)uN,(I−K)v)k+1+hλ,ρviN,n=0, (6)

where uNis the unique solution of (5).

Proof.

For

i=

1,

. . .

,

N

, let us consider

ϕi∈SN

the unique radial basis function determined

by the interpolation conditions

ϕi(tj) = δij,∀j=1, . . . N.

Let take v∈SN, and we consider the function

w=v−

N

∑

i=1

(I−K)v(ti)ϕi,

then

(I−K)w(tj) = (I−k)v(tj)−

N

∑

i=1

(I−K)v(ti)ϕi(tj) = 0, ∀j=1, . . . , N,

that is ρw=0∈RN,n, and in fact w∈H0

N. Thus, from Theorem 2, we have

((I−K)uN,(I−K)w)k+1=0. (7)

We notice

Π`:Rn→R

, for

`=

1,

. . .

,

n

, the projection application given by

Π`(x1, . . . , xn) = x`.

Then, for i=1, . . . , N, it is veriﬁed that

((I−K)uN,(I−K)v(ti)ϕi)k+1=

n

∑

`=1

(Π`((I−K)uN,Π`((I−K)v(ti)ϕi))k+1

=

n

∑

`=1

Π`((I−K)v(ti))(Π`((I−K)uN,ϕi)k+1.

Let denote λi`=−(Π`((I−K)uN,ϕi)k+1∈Rand λ= (λi`)1≤i≤N

1≤`≤n∈RN,n, then

((I−K)uN,(I−K)w)k+1=

((I−K)uN,(I−K)v)k+1−

N

∑

i=1

((I−K)uN,(I−K)v(ti)ϕi)k+1=

((I−K)uN,(I−K)v)k+1+

N

∑

i=1

n

∑

`=1

Π`((I−K)v(ti))λi`=

((I−K)uN,(I−K)v)k+1+hλ,ρviN,n.

From (7), we conclude that there exists

λ= (−(Π`((I−K)un)

,

ϕi)k+1)1≤i≤N

1≤`≤n∈RN,n

such that

((I−K)uN,(I−K)v)k+1+hλ,ρviN,n=0

and (6) holds.

The uniqueness of λis immediate.

Mathematics 2022,10, 223 8 of 15

5. Convergence Result

Assume that

f∈Hk+1((

0, 1

)

;

Rn)

and

k∈Hk((

0, 1

)×(

0, 1

)

;

Rn,n)

, then there exists

a unique solution

x∈Hk+1((

0, 1

)

;

Rn)

of (1). Moreover, the following convergence result

is veriﬁed.

Theorem 4.

Suppose given

f∈Hk+1((

0, 1

)

;

Rn)

and

k∈Hk((

0, 1

)×(

0, 1

)

;

Rn,n)

. Let denote

x∈Hk+1((

0, 1

)

;

Rn)

the unique solution of (1) and

uN∈HN

the unique solution of (4). Suppose

that the hypothesis (2) holds, where h is mentioned. Then, one has

lim

h→0kuN−xkk=0.

Proof.

Let

sx,TN

be the interpolating radial basis function of

x

on

TN

from the Wendland

function φ1,k, then sx,TN∈SN. Thus J(uN)≤J(sx,TN), that also implies that

|(I−K)uN|k+1≤ |(I−K)sx,TN|k+1.

In this case, we have

[[(I−K)uN]] ≤[[(I−K)sx,TN]].

From this, and that the operator

(I−K)

is linear and compact in the ﬁnite-dimensional

space SN, and thus bijective, we can deduce that there exists C1>0 verifying

kuNkk+1≤C1ksx,TNkk+1. (8)

Taking into account (3), it is veriﬁed that there exists C2>0 such

ksx,TNkk+1≤C2kxkk+1.

and, from here and (8) we obtain that there exists C>0 such that

kuNkk+1≤Ckxkk+1.

Thus, the family

(uN)N∈N

is bounded in

Hk+1((

0, 1

)

;

Rn)

, and consequently there

exists a sequence

(uN`)`∈N

extracted from this family, and an element

x∗∈Hk+1((

0, 1

)

;

Rn)

such that

x∗=lim

`→+∞uN`weakly in Hk+1((0, 1);Rn). (9)

Suppose that

x∗6=x

; then, from the continuous injection of

Hk+1((

0, 1

)

;

Rn)

into

C([0, 1];Rn), there exists γ>0 and a nonempty interval ω⊂[0, 1]such that

∀t∈ω,hx∗−xin>γ.

As this injection is compact, from (9)

∃`0∈N,∀`≥`0,huN`(t)−x∗(t)in≤γ

2.

Thus, for any `≥`0and t∈ωit is veriﬁed

huN`(t)−x(t)in≥ hx∗(t)−x(t)in− huN`(t)−x∗(t)in>γ

2. (10)

On the other hand, as we are taking

h→

0 along the whole process, using the density

condition (2) we can assure that there exists

`∈N

and

t∗

`∈ω

such that

t∗

`∈TN`∩ω

and thus

(I−K)uN`(t∗

`) = (I−K)x(t∗

`).

Mathematics 2022,10, 223 9 of 15

The operator

I−K

, considering the hypotheses taken from the beginning, it is also a

bijection in

C((

0, 1

)

;

Rn)

, and thus

uN`(t∗

`) = x(t∗

`)

, which is a contradiction with (10). Thus

x∗=x.

For any `∈Nit is veriﬁed

kuN`−xk2

k=kuN`k2

k+kxk2

k−2(uN`,x)k.

Then, from (9) and the compact inclusion of

Hk+1((

0, 1

)

;

Rn)

into

Hk((

0, 1

)

;

Rn)

(see

for example [48]), one has

lim

`→+∞kuN`−xkk=0. (11)

Suppose now that

kuN−xkk

does not tend to 0 as

h

tends to 0; in this case, it would

exist α>0, and a sequence (uN0

`)`∈Nsuch that

∀`∈N,kuN0

`−xkk>α. (12)

However, the sequence

(uN0

`)`∈N

is bounded in

Hk+1((

0, 1

)

;

Rn)

and then, by reasoning

as above, we deduce that from this sequence we can extract a subsequence convergent to

x

in Hk((0, 1);Rn), what contradicts (12). Thus

lim

h→0kuN−xkk=0.

Corollary 1. Under the conditions of Theorem 4one has

lim

h→0kf−(I−K)uNkk=0.

Proof. From Theorem 4and the continuity of the operator I−Kwe have

lim

h→0(I−K)uN= (I−K)x=fin Hk((0, 1); Rn).

Then, from here the result is obtained.

6. Computation

Let us compute the unique solution of (6). The solution of problem (5) can be ex-

pressed by

uN=

N

∑

i=1

αiφ1,k(|·−ti|),

with α1, . . . , αN∈Rn.

Consider the basis {B1, . . . , BNn}of the space SNgiven, for `=1, . . . Nn, by

B`(t) = φ1,k(|t−ti|)ej,

being i=quotient(`−1, n) + 1 and j=`−(i−1)n.

Then, the solution of (5) can be expressed by

uN=

Nn

∑

`=1

α`B`,

with α1, . . . , αNn ∈R.

Mathematics 2022,10, 223 10 of 15

By replacing in (6), we have

Nn

∑

`=1

α`((I−K)B`,(I−K)v)k+1+hλ,ρviN,n=0, ∀v∈SN,

subject to the restrictions

Nn

∑

`=1

α`(I−K)B`(ti) = f(ti),i=1, . . . , N.

Taking

v=Bj

, for

j=

1,

. . .

,

Nn

, we obtain a linear system of order 2

Nn

with

unknowns α1, . . . , αNn ,λ1, . . . , λNn ∈R, that can be expressed in matrix form as follows:

C D

D>0 α

λ=0

F,

with C=((I−K)B`,(I−K)Bj)k+11≤`≤Nn

1≤j≤Nn ,

D= (dij )1≤i≤Nn

1≤j≤Nn ,

α= (α1, . . . , αNn)>,λ= (λ1, . . . , λN n)>,

F= ( fi)1≤i≤Nn ,

being, for i=1 . . . , Nn and j=1, . . . , Nn,

dij =Π`(( I−K)Br(ts)),

with

r=quotient(i−

1,

n) +

1,

s=quotient(j−

1,

n) +

1,

`=j−(s−

1

)n

and for

i=

1, . . . , Nn,

fi=Π`(f(ts)),

with s=quotient(i−1, n)and `=i−(s−1)n.

7. Numerical Examples

To check the validity of the described method for approximating the solution of

Problem (1) we present some numerical experiments.

In order to show the accuracy of the method, we have computed two relative error

estimations, given by the expressions

E1=1

1000

1000

∑

i=1

hf(ai)−(I−K)uN(ai)in,

which estimates how close uNis to the solution of (1) and

E2=

v

u

u

u

u

u

u

u

t

1000

∑

i=1

huN(ai)−x(ai)i2

n

1000

∑

i=1

hx(ai)i2

n

,

which is an approximation of the relative error of

uN

with respect to

x

in

L2((

0, 1

)

; R

n)

being {a1, . . . , a1000} ⊂ [0, 1]thousand distinct random points.

From Theorem 4and Corollary 1, these relative error estimations

E1

and

E2

tend to 0

as htends to 0.

Mathematics 2022,10, 223 11 of 15

Moreover, in all the examples, the discrete space that we use to calculate the ap-

proximated solution

uN

is the radial basis function space constructed from the Wendland

function φ1,1 and the centers set TN={ti=i

N,i=0, . . . , N}.

In order to compute the numerical integrals, we have used the following quadrature

formula (see [52])

Zb

ag(t)dt ≈

n−3

∑

i=6

g(ξi) + h206

1575(g(ξ1) + g(ξn+2)) + 107

128(g(ξ2) + g(ξn+1))+

6019

5760(g(ξ3) + g(ξn)) + 9467

9600(g(ξ4) + g(ξn−1))+

13,469

13,440(g(ξ5) + g(ξn−2)),

where h=b−a

nand

ξ1=a,ξn+2=b,ξi=a+2i−1

2h,i=2, . . . , n+1.

This formula has an error order of O(h6)for g∈C6([a,b]).

Example 1. We consider the following Volterra equation system of order 2

x1(t)−Zt

0((t−s)3x1(s) + (t−s)2x2(s))ds =t−t5

12,

x2(t)−Zt

0((t−s)4x1(s) + (t−s)3x2(s))ds =t2−t6

20.

The exact solution is

x1(t) = t,x2(t) = t2.

Table 2shows the relative error estimations for distinct values of N.

Table 2. Computed relative error estimations for Example 1from some values of N.

N E1E2

52.1868 ×10−23.1058 ×10−2

10 3.6034 ×10−34.8048 ×10−3

20 6.2683 ×10−48.2990 ×10−4

30 2.0727 ×10−43.0254 ×10−4

40 1.0215 ×10−41.2509 ×10−4

50 6.4520 ×10−59.2824 ×10−5

Example 2. We consider the following Volterra equation system of order 2

x1(t)−Zt

0(et−sx1(s) + et+sx2(s))ds =et(1−2t),

x2(t)−Zt

0(−et−sx1(s) + et+sx2(s))ds =e−t.

The exact solution is

x1(t) = et,x2(t) = e−t.

Table 3shows the relative error estimations for distinct values of N.

Mathematics 2022,10, 223 12 of 15

Table 3. Computed relative error estimations for Example 2from some values of N.

N E1E2

53.0586 ×10−22.5854 ×10−2

10 6.4473 ×10−33.7229 ×10−3

20 1.1610 ×10−36.3689 ×10−4

30 4.4048 ×10−42.2905 ×10−4

40 1.5159 ×10−41.1068 ×10−4

50 9.9079 ×10−56.3629 ×10−5

Example 3. We consider the following Volterra equation system of order 3

x1(t)−Zt

0(x1(s) + tx3(s))ds =−t+t2,

x2(t)−Zt

0((t+s)x1(s) + x2(s) + (t−s)x3(s))ds =1−t−t4

2,

x3(t)−Zt

0((−t−s)x1(s)−x2(s) + (−t+s)x3(s))ds =−t−t2+t4

2.

The exact solution is

x1(t) = t2,x2(t) = 1, x3(t) = −t2.

Table 4shows the relative error estimations for distinct values of N.

Table 4. Computed relative error estimations for Example 3from some values of N.

N E1E2

52.0024 ×10−23.5705 ×10−2

10 2.4457 ×10−35.2296 ×10−3

20 2.9878 ×10−47.5222 ×10−4

30 7.5462 ×10−52.6518 ×10−4

40 2.6133 ×10−51.1834 ×10−4

50 1.0932 ×10−57.1453 ×10−5

8. Conclusions

We conclude that the above presented experiments (see Tables 1–4)) conﬁrm the valid-

ity of the method and justify the convergence results given in Theorem 4and Corollary 1. In

fact, in all our experiments (see the Examples 1–3), by using small values of

N

, we obtain a

signiﬁcant good order of approximation using the relative errors

E1

and

E2

considered. So,

our original goal to devise an appropriate variational procedure that is capable of solving

this type of problems in a precise and efﬁcient way has been completely accomplished.

As compared with the other recently published works, for example [

36

–

38

,

40

,

41

], they

do not study convergence results. Likewise, our technique gives an acceptable accuracy

with a small use of data, resulting also a low computational cost.

In ([

37

], Tables 1 and 2) the mean of the error is of the order 10

−5

. We have obtained

the same order of error with only 50 points.

In [

40

] the authors use Bernstein polynomials and the degree of its approximation is

of order 10

−4

in most of the tables. The same happens in ([

41

], Table 4), it uses the simple

block-by-block method and its degree of approximation is about 10−3.

In order to do more research on this topic in the future, among some of the open

problems that we consider are:

Mathematics 2022,10, 223 13 of 15

– a numerical comparison between our method and many others in the literature,

– the theoretical study of the order of convergence of the presented method,

–

the adaptation of this procedure to ﬁnd the numerical solution of the linear systems of

2D Volterra integral equations of the second kind.

Author Contributions:

All authors contributed equally to formal analysis, investigation, methodol-

ogy, project administration, writing—original draft and writing—review & editing. All authors have

read and agreed to the published version of the manuscript.

Funding:

This work was supported by FEDER/Junta de Andalucía-Consejería de Transformación

Económica, Industria, Conocimiento y Universidades (Research Project A-FQM-76-UGR20, University

of Granada) and by the Junta de Andalucía (Research Group FQM191).

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments:

We acknowledge the anonymous referees for their useful comments and sug-

gestions, as well as the Department of Applied Mathematics of the University of Granada and the

Mathematics Journal Editorial Board, for the ﬁnancial aid offered for the ﬁnal cost of the APC.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Volterra, V. Leçons sur les équations intégrales et intégro-différentielles; Gauthier-Villars: Paris, France, 1913.

2. Corduneanu, C.; Sandberg, I.W. Volterra Equations and Applications; Gordon and Breach: Amsterdam, The Netherlands, 2000.

3.

Baker, C.T.H. Numerical Analysis of Volterra Functional and Integral Equations. In The State of the Art in Numerical Analysis;

Duff, I.S., Watson, G.A., Eds.; Oxford University Press: New York, NY, USA, 1997; pp. 193–222.

4. Cahlon, B. Numerical solution of nonlinear Volterra integral equations. J. Comput. Appl. Math. 1981,7, 121–128. [CrossRef]

5. Shampine, L.F. Solving Volterra integral equations with ODE codes. IMA J. Numer. Anal. 1988,8, 37–41. [CrossRef]

6.

Brunner, H. 1896–1996: One hundred years of Volterra integral equations of the ﬁrst kind. Appl. Numer. Math.

1997

,24, 83–93.

[CrossRef]

7.

Brunner, H. ; Pedas, A.; Vainikko, G. The piecewise polynomial collocation method for nonlinear weakly singular Volterra

equations. Math. Comp. 1999,68, 1079–1095. [CrossRef]

8.

Elnagar, G.N.; Kazemi, M. Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations. J. Comput.

Appl. Math. 1996,76, 147–158. [CrossRef]

9.

Daubechies I. Ten Lectures on Wavelets; CBMS-NSF Regional Conference Series in Applied Mathematics; SIAM: Philadelphia, PA,

USA, 1992.

10.

Aziz, I.; Siraj-ul-Islam. New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using

Haar wavelets. J. Comput. Appl. Math. 2013,239, 333–345. [CrossRef]

11.

Babolian, E.; Shahsavaran, A. Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar

wavelets. J. Comput. Appl. Math. 2009,225, 87–95. [CrossRef]

12.

Lepik, Ü. Solving integral and differential equations by the aid of non-uniform Haar wavelets. Appl. Math. Comput.

2008

,198,

326–332. [CrossRef]

13.

Maleknejad, K.; Mirzaee, F. Using rationalized Haar wavelet for solving linear integral equations. Appl. Math. Comput.

2005

,160,

579–587. [CrossRef]

14.

Mirzaae, F. Numerical computational solution of the linear Volterra integral equations systems via rationalized Haar functions.

J. King Saud Univ. Sci. 2010,22, 265–268. [CrossRef]

15.

Maleknejad, K.; Almasieh, H.; Roodaki, M. Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm

integral equations. Comm. Nonlinear Sci. Numer. Simulat. 2010,15, 3293–3298. [CrossRef]

16.

Deb, A.; Sarkar, G.; Sengupta, A. Triangular Orthogonal Functions for the Analysis of Continuous Time Systems; Elsevier: New Delhi,

India, 2007.

17.

Roodaki, M.; Almasieh, H. Delta basis functions and their applications to systems of integral equations. Comput. Math. Appl.

2012,63, 100–109. [CrossRef]

18.

Ghasemi, M.; Tavassoli Kajani, M.; Babolian, E. Numerical Solutions of the Nonlinear Volterra-Fredholm Integral Equations by

Using Homotopy Perturbation Method. Appl. Math. Comput. 2007,188, 446–449. [CrossRef]

19.

Wazwaz, A.-M. The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterra integro–

differential equations. Appl. Math. Comput. 2010,216, 1304–1309. [CrossRef]

20.

Youseﬁ, S.; Razzaghi, M. Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations. Math. Comput.

Simulat. 2005,70, 1–8. [CrossRef]

Mathematics 2022,10, 223 14 of 15

21.

Maleknejad, K.; Kajani, M.T. Solving integro-differential equation by using Hybrid Legendre and block-pulse functions. Int. J.

Appl. Math. 2002,11, 67–76.

22.

El-Borai, M.; Abdou, M.A.; Badr, A.A.; Basseem, M. Chebyshev Polynomials and Fredholm-Volterra integral equation. Int. J. Appl.

Math. Mech. 2008,4, 78–92.

23.

Youssri, Y.H.; Hafez, R. M. Chebyshev collocation treatment of Volterra–Fredholm integral equation with error analysis.

Arab. J. Math. 2020,9, 471–480. [CrossRef]

24.

Berenguer, M.I.; Gámez, D. ; Garralda-Guillém, A.I. ; Ruiz Galán, M. ; Serrano Pérez, M.C. Analytical Techniques for a Numerical

Solution of the Linear Volterra Integral Equation of the Second Kind. Abstr. Appl. Anal. 2009, 14936. [CrossRef]

25.

Berenguer, M.I.; Gámez, D. ; Garralda-Guillém, A.I. ; Ruiz Galán, M. ; Serrano Pérez, M.C. Nonlinear Volterra Integral Equation

of the Second Kind and Biorthogonal Systems. Abstr. Appl. Anal. 2010, 135216. [CrossRef]

26.

¸Sahn, N.; Yüzba¸si, ¸S.; Glsu, M. A collocation approach for solving systems of linear Volterra integral equations with variable

coefﬁcients. Comput. Math. Appl. 2011,62, 755–769. [CrossRef]

27.

Balakumar, V.; Murugesan, K. Numerical solution of Volterra integral-algebraic equations using block pulse functions.

Appl. Math. Comput. 2015,263, 165–170. [CrossRef]

28.

Yang, L.-H.; Shen, J.-H.; Wang, Y. The reproducing kernel method for solving the system of the linear Volterra integral equations

with variable coefﬁcients. J. Comput. Appl. Math. 2012,236, 2398–2405. [CrossRef]

29. Delves, L.M.; Mohamed, J.L. Computational Methods for Integral Equations; Cambridge University Press: New York, USA, 1985.

30.

Van der Houwen, P.J.; Sommeijer, B.P. Euler-Chebyshev methods for integro-differential equations. Appl. Numer. Math.

1997

,24,

203–218. [CrossRef]

31.

Draidi, W.; Qatanani, N. Numerical Schemes for solving Volterra integral equations with Carleman kernel. Int. J. Appl. Math.

2005,31, 647–669. [CrossRef]

32.

Issa, A.; Qatanani, N.; Daraghmeh, A. Approximation Techniques for Solving Linear Systems of Volterra Integro-Differential

Equations. J. Appl. Math. 2020, 1–12. [CrossRef]

33.

Aggarwal, S.; Gupta, A.R. Solution of Linear Volterra Integro-Differential Equations of Second Kind Using Kamal Transform.

J. Emerg. Technol. Innov. Res. 2019,6, 741–747.

34.

Chauhan, R.; Aggarwal, S. Laplace Transform for Convolution Type Linear Volterra Integral Equation of Second Kind. J. Adv. Res.

Appl. Math. Stat. 2019,4, 1–7.

35. Mahgoub, M.M.A. The new integral transform: Sawi transform. Adv. Theor. Appl. Math. 2019,14, 81–87.

36.

Islam, M.S.; Islam, M.S.; Bangalee, M.Z.I.; Khan, A.K.; Halder, A. Approximate Solution of Systems of Volterra Integral Equations

of Second Kind by Adomian Decomposition Method. Dhaka. Univ. J. Sci. 2015 63, 15–18. [CrossRef]

37.

Hong, Z.; Fang, X.; Yan, Z., Hao, H. On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method.

J. Appl. Math. Phys. 2016,4, 1315–1320. [CrossRef]

38.

Khan, F.; Omar, M.; Ullah, Z. Discretization method for the numerical solution of 2D Volterra integral equation based on

two-dimensional Bernstein polynomials. AIP Adv. 2018,8, 125209. [CrossRef]

39.

Aggarwal, S.; Sharma, N.; Chauhan, R. Solution of Linear Volterra Integral Equations of Second Kind Using Mohand Transform.

IJRAT 2018,6, 3098–3102.

40.

Muhammad, A.M.; Ayal, A.M. Numerical Solution of Linear Volterra Integral Equation with Delay using Bernstein Polynomial.

IEJME 2019,14, 735–740. [CrossRef]

41.

Kasumo, C. On the Approximate Solutions of Linear Volterra Integral Equations of the First Kind. Appl. Math. Sci.

2020

,14,

141–153. [CrossRef]

42.

Bjornsson, H.; Hafstein, S. Advanced algorithm for interpolation with Wendland functions. In Informatics in Control, Automation

and Robotics (ICINCO 2019); Lecture Notes in Electrical Engineering; Gusikhin O., Madani K., Zaytoon J., Eds.; Springer: New York,

NY, USA, 2019; pp. 99–117.

43.

Assari, P.; Dehghan, M. A meshless local Galerkin method for solving Volterra integral equations deduced from nonlinear

fractional differential equations using the moving least squares technique. Appl. Numer. Math. 2019,143, 276–299. [CrossRef]

44.

Farshadmoghadam, F.; Azodi, H.D. ; Yaghouti, M.R. An improved radial basis functions method for the high-order Volterra-

Fredholm integro-differential equations. Math. Sci. 2021. [CrossRef]

45.

Maleknejad, K.; Mohammadikia, H.; Rashidinia, J. A numerical method for solving a system of Volterra–Fredholm integral

equations of the second kind based on the meshless method. Afrika Mat. 2018,29, 955–965. [CrossRef]

46.

Uddin, M.; Ullah, N.; Ali-Shah, S.I. RBF Based Localized Method for Solving Nonlinear Partial Integro-Differential Equations.

Comput. Model. Eng. Sci. 2020,123, 957–972. [CrossRef]

47.

Zhang, H.; Chen, Y.; Nie, X. Solving the Linear Integral Equations Based on Radial Basis Function Interpolation. J. Appl. Math.

2014

, 1–18. Available online: https://www.hindawi.com/journals/jam/2014/793582/ (accessed on 20 October 2021). [CrossRef]

48.

Atkinson, K.; Weiman, H. Theoretical Numerical Analysis. A Functional Analysis Framework, 2nd ed.; Springer: New York, NY,

USA, 2005.

49.

Kouibia, A.; Pasadas, M.; Rodriguez, M.L. A variational method for solving Fredholm integral systems. Appl. Numer. Math.

2012

,

66, 1041–1049. [CrossRef]

50.

Wendland, H. Error Estimates for Interpolation by Compactly Supported Radial Basis Functions of Minimal Degree. J. Approx.

Theory 1998,93, 258–272. [CrossRef]