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Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

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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.
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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
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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 st1, (1)
where
x(t) = (x1(t), . . . , xn(t))>,
f(t) = ( f1(t), . . . , fn(t))>,
k(t,s) = (kij(t,s))1i,jn.
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= (IK)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={xR: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
uL2((
0, 1
)
;
Rn)
together with all
j
-th derivative functions
u(j)of order jm, in the sense of distributions. This space is equipped with
the semi–inner products, for any u,vHm((0, 1);Rn),
(u,v)j=Z1
0hu(j)(t),v(j)(t)indt, 0 jm,
the corresponding semi–norms |u|j= (u,u)1
2
j, for 0 jm,
the inner product ((u,v))m=
m
0
(u,v)j,
and the corresponding norm kukm= ((u,u)) 1
2
m.
For any 1
in
, 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))1inHm((0, 1)×(0, 1);Rn,n),
together with the associated integral operator
Ku(t) = Zt
0hki(t,s),u(s)inds1in
,t(0, 1),uHm((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 )1in
1jp,B= (bij )1in
1jpRn,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+
0R
, a subset
Rd
,
d
1, and a point
ξ
, the radial function defined on
from the function
φ
with center
ξ
is the continuous function
Φξ:R given by
Φξ(x) = φ(hxξid).
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
uC([
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(|tti|),t[0, 1],
where
φ:R+
0R
is a continuous function and the coefficients
α1
,
. . .
,
αNRn
are determined
by the interpolation conditions
su,TN(ti) = u(ti), 1 iN.
In [
50
] H. Wendland introduced a family of compactly supported radial basis functions
in the following way: let the operator Iand its inverse Dfor r0 be given by
(Iφ)(r) = Z
rtφ(t)dt,
(Dφ)(r) = 1
rφ0(r),
for any differentiable function φ:R+
0R.
Given the truncated power function φ`(r) = (1r)`
+, 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 r1,
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
kN
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) = (1r)+C0
k=1φ1,1(r).
= (1r)3
+(3r+1)C2
k=2φ1,2(r).
= (1r)5
+(8r2+5r+1)C4
Let
h=sup
t[0,1]
min
1iN|tti|. (2)
From ([
50
], Theorem 2.1) we can affirm that
φ1,kC2k([
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
kusu,TNkL((0,1);Rn)Ckukk+1hk+1
2,uHk+1([0, 1];Rn),
and
|usu,TN|jChk+1jkukk+1, 0 jk+1, uHk+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 SNHk+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= ((IK)v(ti))1iN.
Let assume that
fHk+1((
0, 1
)
;
Rn)
and consider the affine variety
HN={uSN:
ρu= ( f(ti))1iN}and the linear subspace H0
N={uSN:ρu=0RN,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+ ((IK)u,(IK)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
uNHN
is an approximating radial basis function relative to
TN
,
ρ
and fif uNis a solution of the following minimization problem:
Find uNHNsuch that vHN,J(uN)J(v), (4)
where J :Hk+1((0, 1);Rn)Ris given by
J(v) = |(IK)v|2
k+1.
Theorem 2.
Problem (4) has a unique solution
uNHN
which is the unique solution of the
variational problem
vH0
N,((IK)uN,(IK)v)k+1=0. (5)
Proof.
From Proposition 1and ([
48
], Theorem 3.4.3) we can deduce that there exists a
unique uNHN, which is the projection of 0on HNsuch that
[[uN]] [[v]],vHN
and verifying
wHN,<< uN,wuN>>0,
that is
vH0
N,<< uN,v>>0
Mathematics 2022,10, 223 7 of 15
and, taking into account that H0
Nis a vector space, we obtain that
vH0
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 vHN.
Theorem 3. There exists a unique λRN,nsuch that
vSN,((IK)uN,(IK)v)k+1+hλ,ρviN,n=0, (6)
where uNis the unique solution of (5).
Proof.
For
i=
1,
. . .
,
N
, let us consider
ϕiSN
the unique radial basis function determined
by the interpolation conditions
ϕi(tj) = δij,j=1, . . . N.
Let take vSN, and we consider the function
w=v
N
i=1
(IK)v(ti)ϕi,
then
(IK)w(tj) = (Ik)v(tj)
N
i=1
(IK)v(ti)ϕi(tj) = 0, j=1, . . . , N,
that is ρw=0RN,n, and in fact wH0
N. Thus, from Theorem 2, we have
((IK)uN,(IK)w)k+1=0. (7)
We notice
Π`:RnR
, for
`=
1,
. . .
,
n
, the projection application given by
Π`(x1, . . . , xn) = x`.
Then, for i=1, . . . , N, it is verified that
((IK)uN,(IK)v(ti)ϕi)k+1=
n
`=1
(Π`((IK)uN,Π`((IK)v(ti)ϕi))k+1
=
n
`=1
Π`((IK)v(ti))(Π`((IK)uN,ϕi)k+1.
Let denote λi`=(Π`((IK)uN,ϕi)k+1Rand λ= (λi`)1iN
1`nRN,n, then
((IK)uN,(IK)w)k+1=
((IK)uN,(IK)v)k+1
N
i=1
((IK)uN,(IK)v(ti)ϕi)k+1=
((IK)uN,(IK)v)k+1+
N
i=1
n
`=1
Π`((IK)v(ti))λi`=
((IK)uN,(IK)v)k+1+hλ,ρviN,n.
From (7), we conclude that there exists
λ= ((Π`((IK)un)
,
ϕi)k+1)1iN
1`nRN,n
such that
((IK)uN,(IK)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
fHk+1((
0, 1
)
;
Rn)
and
kHk((
0, 1
)×(
0, 1
)
;
Rn,n)
, then there exists
a unique solution
xHk+1((
0, 1
)
;
Rn)
of (1). Moreover, the following convergence result
is verified.
Theorem 4.
Suppose given
fHk+1((
0, 1
)
;
Rn)
and
kHk((
0, 1
)×(
0, 1
)
;
Rn,n)
. Let denote
xHk+1((
0, 1
)
;
Rn)
the unique solution of (1) and
uNHN
the unique solution of (4). Suppose
that the hypothesis (2) holds, where h is mentioned. Then, one has
lim
h0kuNxkk=0.
Proof.
Let
sx,TN
be the interpolating radial basis function of
x
on
TN
from the Wendland
function φ1,k, then sx,TNSN. Thus J(uN)J(sx,TN), that also implies that
|(IK)uN|k+1≤ |(IK)sx,TN|k+1.
In this case, we have
[[(IK)uN]] [[(IK)sx,TN]].
From this, and that the operator
(IK)
is linear and compact in the finite-dimensional
space SN, and thus bijective, we can deduce that there exists C1>0 verifying
kuNkk+1C1ksx,TNkk+1. (8)
Taking into account (3), it is verified that there exists C2>0 such
ksx,TNkk+1C2kxkk+1.
and, from here and (8) we obtain that there exists C>0 such that
kuNkk+1Ckxkk+1.
Thus, the family
(uN)NN
is bounded in
Hk+1((
0, 1
)
;
Rn)
, and consequently there
exists a sequence
(uN`)`N
extracted from this family, and an element
xHk+1((
0, 1
)
;
Rn)
such that
x=lim
`+uN`weakly in Hk+1((0, 1);Rn). (9)
Suppose that
x6=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
tω,hxxin>γ.
As this injection is compact, from (9)
`0N,``0,huN`(t)x(t)inγ
2.
Thus, for any ``0and tωit 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
(IK)uN`(t
`) = (IK)x(t
`).
Mathematics 2022,10, 223 9 of 15
The operator
IK
, 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
k2(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
kuNxkk
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
h0kuNxkk=0.
Corollary 1. Under the conditions of Theorem 4one has
lim
h0kf(IK)uNkk=0.
Proof. From Theorem 4and the continuity of the operator IKwe have
lim
h0(IK)uN= (IK)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, . . . , αNRn.
Consider the basis {B1, . . . , BNn}of the space SNgiven, for `=1, . . . Nn, by
B`(t) = φ1,k(|tti|)ej,
being i=quotient(`1, n) + 1 and j=`(i1)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`((IK)B`,(IK)v)k+1+hλ,ρviN,n=0, vSN,
subject to the restrictions
Nn
`=1
α`(IK)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=((IK)B`,(IK)Bj)k+11`Nn
1jNn ,
D= (dij )1iNn
1jNn ,
α= (α1, . . . , αNn)>,λ= (λ1, . . . , λN n)>,
F= ( fi)1iNn ,
being, for i=1 . . . , Nn and j=1, . . . , Nn,
dij =Π`(( IK)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(i1, n)and `=i(s1)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)(IK)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
n3
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(ξn1))+
13,469
13,440(g(ξ5) + g(ξn2)),
where h=ba
nand
ξ1=a,ξn+2=b,ξi=a+2i1
2h,i=2, . . . , n+1.
This formula has an error order of O(h6)for gC6([a,b]).
Example 1. We consider the following Volterra equation system of order 2
x1(t)Zt
0((ts)3x1(s) + (ts)2x2(s))ds =tt5
12,
x2(t)Zt
0((ts)4x1(s) + (ts)3x2(s))ds =t2t6
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 ×1023.1058 ×102
10 3.6034 ×1034.8048 ×103
20 6.2683 ×1048.2990 ×104
30 2.0727 ×1043.0254 ×104
40 1.0215 ×1041.2509 ×104
50 6.4520 ×1059.2824 ×105
Example 2. We consider the following Volterra equation system of order 2
x1(t)Zt
0(etsx1(s) + et+sx2(s))ds =et(12t),
x2(t)Zt
0(etsx1(s) + et+sx2(s))ds =et.
The exact solution is
x1(t) = et,x2(t) = et.
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 ×1022.5854 ×102
10 6.4473 ×1033.7229 ×103
20 1.1610 ×1036.3689 ×104
30 4.4048 ×1042.2905 ×104
40 1.5159 ×1041.1068 ×104
50 9.9079 ×1056.3629 ×105
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) + (ts)x3(s))ds =1tt4
2,
x3(t)Zt
0((ts)x1(s)x2(s) + (t+s)x3(s))ds =tt2+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 ×1023.5705 ×102
10 2.4457 ×1035.2296 ×103
20 2.9878 ×1047.5222 ×104
30 7.5462 ×1052.6518 ×104
40 2.6133 ×1051.1834 ×104
50 1.0932 ×1057.1453 ×105
8. Conclusions
We conclude that the above presented experiments (see Tables 14)) 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 13), 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 103.
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.
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