Fixed-Point Iterative Algorithm for the Linear Fredholm-Volterra Integro-Differential Equation

Abstract

With the aid of fixed-point theorem (an equivalent version for the linear case) and biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a linear Fredholm-Volterra integro-differential equation is turned into a numerical algorithm, so that it can be solved numerically.

1. Introduction

Denoting by 𝒞([0,1]) and 𝒞([0,1]²) the Banach space of all continuous and real-valued functions defined on [0,1] and [0,1]², respectively, equipped with their usual sup-sup norm, let us consider the following problem associated to the Fredholm-Volterra integro-differential equation: given 𝜌∈ℝ,𝑘₁,𝑘₂∈𝒞([0,1]²), and 𝑓,𝑥₀∈𝒞([0,1]), find 𝑥∈𝒞¹([0,1]) such that 𝑥^′(𝑡)=𝑥₀(𝑡)+𝑓(𝑡)𝑥(𝑡)+¹₀𝑘₁(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+^𝑡₀𝑘₂(𝑡,𝑠)𝑥(𝑠)𝑑𝑠(𝑡,𝑠)∈[0,1]²,𝑥(0)=𝜌. (1.1)

Observe that if 𝑘₁(𝑡,𝑠)=0 and 𝑓(𝑡)=0 in (1.1), the equation is transformed into a linear Volterra integro-differential equation; and if 𝑘₂(𝑡,𝑠)=0 and 𝑓(𝑡)=0, it becomes a linear Fredholm integro-differential. Additionally, if 𝑘₁(𝑡,𝑠)=𝑘₂(𝑡,𝑠)=0, (1.1) is transformed into a linear differential equation of the first order.

Frequently the mathematical modelling of problems arising from the real world (see [1] and the references therein) deals with problem (1.1). These are usually difficult to solve analytically, and in many cases, the solution must be approximated. Therefore, in recent years, several numerical approaches have been proposed (see, e.g., [2–4]). The numerical methods usually transform the integro-differential equation into a linear system that can be solved by direct or iterative methods. On the other hand, the use of fixed-point techniques in the numerical study of linear differential, integral, and integro-differential equations has also proven successful in some works, as [5–11]. The purpose of this paper is to develop an effective method for approximating the solution of (1.1) using Schauder basis and another classical tool in analysis: an equivalent version of the fixed-point theorem for the linear case (the geometric series theorem). This algorithm generalizes the developed ones in [7, 8, 10] for Volterra integro-differential, Fredholm integro-differential, and differential equation, respectively.

To establish our numerical method, we first need to review some results of a theoretical nature in Section 2. We arrive at a numerical method for approximating the solution of (1.1) in Section 3, and in order to state the results about convergence and to study the error of the proposed algorithm, we will assume that 𝛾=‖𝑓‖+‖𝑘₁‖+‖𝑘₂‖/2<1 and 𝑘₁,𝑘₂∈𝐶¹([0,1]²) and 𝑓,𝑔∈𝐶¹([0,1]). Finally, in Section 4, we illustrate the theoretical results with two examples.

2. Two Tools of a Theoretical Nature

Two fundamental tools will be used to establish the algorithm needed to solve the problem (1.1). The first is the following equivalent version (for the linear case) of the Banach fixed-point theorem (see [12]).

Geometric Series Theorem
Let 𝑋 be a Banach space, and let 𝑇∶𝑋→𝑋 be a continuous and linear operator such that ‖𝑇‖<1. Then, 𝐼𝑑−𝑇 is a continuous, linear, and bijective operator and (𝐼𝑑−𝑇)⁻¹=∑_𝑛≥0𝑇^𝑛.

In particular, given 𝑦∈𝑋, the equation (𝐼𝑑−𝑇)𝑥=𝑦 has a unique solution 𝑥=(𝐼𝑑−𝑇)⁻¹𝑦.

The second tool applied consists of biorthogonal systems in Banach spaces 𝒞([0,1]) and 𝒞([0,1]²). We will make use of the usual Schauder basis for simplicity in the exposition, although the numerical method given works equally well by replacing it with any complete biorthogonal system in 𝒞([0,1]²). For this reason, we will now briefly recall the main issues and notations regarding Schauder bases in 𝒞([0,1]) and 𝒞([0,1]²).

Let us consider the usual Schauder basis {𝑏_𝑛}_𝑛≥1 in the space 𝒞([0,1]), that is, for a dense sequence {𝑡_𝑖}_𝑖≥1 of distinct real numbers in [0,1] such that 𝑡₁=0 and 𝑡₂=1, we define 𝑏₁(𝑡)∶=1 for 𝑡∈[0,1], and for 𝑛≥2, 𝑏_𝑛 is the piecewise linear continuous function on [0,1] with nodes at {𝑡_𝑖∶1≤𝑖≤𝑛}, uniquely determined by the relations 𝑏_𝑛(𝑡_𝑛)=1 and 𝑏_𝑛(𝑡_𝑘)=0 for 𝑘<𝑛. For each function 𝑧∈𝒞([0,1]), there exists a unique sequence of scalars {𝛽_𝑛}_𝑛≥1 such that 𝑧=∑_𝑛≥1𝛽_𝑛𝑏_𝑛. We denote by {𝑏^∗_𝑛}_𝑛≥1 the sequence of (continuous and linear) biorthogonal functionals in the dual space of 𝒞([0,1]), defined by 𝑏^∗_𝑛𝑘≥1𝛽_𝑘𝑏_𝑘=𝛽_𝑛, (2.1) whereas the sequence of (continuous and linear) projections {𝑃_𝑛}_𝑛≥1 is defined by the partial sums 𝑃_𝑛𝑘≥1𝛽_𝑘𝑏_𝑘=^𝑛_𝑘=1𝛽_𝑘𝑏_𝑘. (2.2)

For the respective sequence of biorthogonal systems {𝑏^∗_𝑛}_𝑛≥1 and each 𝑧∈𝒞([0,1]), the equalities 𝑏^∗₁(𝑧)=𝑧𝑡₁,𝑏^∗_𝑛(𝑧)=𝑧𝑡_𝑛−^𝑛−1_𝑘=1𝑏^∗_𝑘(𝑧)𝑏_𝑘𝑡_𝑛,for𝑛≥2 (2.3) are valid (see [13]). As a consequence, the sequence of associated projections {𝑃_𝑛}_𝑛≥1 satisfies the following interpolation property: 𝑧∈𝒞([0,1]),𝑛≥1,𝑖≤𝑛⟹𝑃_𝑛(𝑧)𝑡_𝑖=𝑧𝑡_𝑖. (2.4)

We now evoke the construction of the usual Schauder basis {𝐵_𝑛}_𝑛≥1 for the Banach space 𝒞([0,1]²), endowed with its usual sup-norm (see [13, 14]). To this end, we consider the bijective mapping 𝜏∶ℕ→ℕ×ℕ (where [𝑝] denotes the integer part of 𝑝) given by 𝜏(𝑛)√∶=𝑛,√𝑛if√𝑛=√√𝑛,𝑛−𝑛²,√𝑛+1if0<𝑛−√𝑛²≤√𝑛,√𝑛+1,𝑛−√𝑛²−√𝑛if√𝑛<𝑛−√𝑛². (2.5) Then it suffices to define 𝐵_𝑛(𝑡,𝑠)∶=𝑏_𝑖(𝑡)𝑏_𝑗(𝑠),𝑡,𝑠∈[0,1], (2.6) whenever 𝜏(𝑛)=(𝑖,𝑗). Let {𝐵^∗_𝑛}_𝑛≥1 and {𝑄_𝑛}_𝑛≥1 stand for the corresponding sequences of biorthogonal functionals and projections, respectively. The Schauder basis {𝐵_𝑛}_𝑛≥1 has similar properties to those of the one-dimensional basis {𝑏_𝑛}_𝑛≥1: (a)for all 𝑡,𝑠∈[0,1], 𝐵₁(𝑡,𝑠)=1 and for 𝑛≥2, 𝐵_𝑛(𝑡_𝑖,𝑡_𝑗)=1if𝜏(𝑛)=(𝑖,𝑗),0if𝜏⁻¹(𝑖,𝑗)<𝑛,(b)if 𝑤∈𝐶([0,1]²), then 𝐵^∗₁(𝑤)=𝑤(𝑡₁,𝑡₁), and for all 𝑛≥2, if 𝜏(𝑛)=(𝑖,𝑗), 𝐵^∗_𝑛(𝑤)=𝑤(𝑡_𝑖,𝑡_𝑗)−∑^𝑛−1_𝑘=1𝐵^∗_𝑘(𝑤)𝐵_𝑘(𝑡_𝑖,𝑡_𝑗),(c)the sequence of associated projections {𝑄_𝑛}_𝑛≥1 satisfies 𝑄_𝑛(𝑤)(𝑡_𝑖,𝑡_𝑗)=𝑤(𝑡_𝑖,𝑡_𝑗), whenever 𝑛,𝑖,𝑗∈ℕ and 𝜏⁻¹(𝑖,𝑗)≤𝑛,(d)this Schauder basis is monotone, that is, sup{‖𝑄_𝑛‖}_𝑛∈ℕ=1.

Observe that for each 𝑛∈ℕ, the definition of the projections 𝑃_𝑛 and 𝑄_𝑛2 requires only the first 𝑛+1 points of the sequence {𝑡_𝑖}_𝑖≥1. These 𝑛+1 points, sorted in increasing order, constitute a partition of the interval [0,1], which will be denoted as 𝑇_𝑛; let Δ𝑇_𝑛 denote the maximum distance between two consecutive points of 𝑇_𝑛. Under some weak conditions, from (2.4) and (c), respectively, and the mean-value theorem, we can estimate the rate of convergence of the sequence of projections: 𝑧∈𝐶¹([0,1]),𝑛≥2⟹𝑧−𝑃_𝑛(𝑧)≤2𝑧′Δ𝑇_𝑛.𝑤∈𝐶¹[0,1]²𝑛≥2⟹𝑤−𝑄_𝑛2(𝑤)≤4max𝜕𝑤,𝜕𝑡𝜕𝑤𝜕𝑠Δ𝑇_𝑛. (2.7)

3. The Algorithm: Convergence and Error

Our starting point is the formulation of (1.1) in terms of a certain operator 𝐿 as follows: let 𝐿∶𝒞([0,1])→𝒞([0,1]) be the linear and continuous operator defined by 𝐿𝑥(𝑡)∶=^𝑡₀𝑓(𝑢)𝑥(𝑢)𝑑𝑢+^𝑡₀¹₀𝑘₁()𝑥(𝑠)+𝑢,𝑠𝑑𝑠𝑑𝑢_𝑡0𝑢0𝑘₂(𝑢,𝑠)𝑥(𝑠)𝑑𝑠𝑑𝑢(0≤𝑡≤1,𝑥∈𝒞([0,1])). (3.1)

It is a simple matter to check that a function 𝑥∈𝐶¹([0,1]) is the solution of (1.1) if and only if (𝐼𝑑−𝐿)𝑥=𝑔, where 𝑔(𝑡)∶=𝜌+∫^𝑡₀𝑥₀(𝑢)𝑑𝑢.

It can be shown by an induction argument and Fubini's theorem that for 𝑗∈ℕ, ‖𝐿^𝑗𝑥‖≤‖𝑥‖𝛾^𝑗; hence, _𝑗≥0𝐿^𝑗𝑥≤‖𝑥‖_𝑗≥0𝛾^𝑗=‖𝑥‖.1−𝛾 (3.2) The condition assumed on 𝛾 and the geometric series theorem allow us to establish the existence of one and only one solution 𝑥 of (1.1), which is 𝑥=_𝑗≥0𝐿^𝑗𝑔. (3.3) Besides, 𝑥−^𝑚_𝑗=0𝐿^𝑗𝑔=_{𝑗≥𝑚+1}𝐿^𝑗𝑔≤_{𝑗≥𝑚+1}𝐿^𝑗𝑔≤𝑔_{𝑗≥𝑚+1}𝛾^𝑗=𝛾^𝑚+1𝑔1−𝛾. (3.4)

In view of (3.3), we consider the sequence {𝑦_𝑚}_𝑚≥0 defined as follows: 𝑦₀(𝑡)=𝐿⁰𝑔(𝑡)=𝑔(𝑡), and for 𝑚≥1, 𝑦_𝑚(𝑡)=^𝑚_𝑗=0𝐿^𝑗𝑔(𝑡)=𝑔(𝑡)+𝐿𝑦_𝑚−1(𝑡)=𝑔(𝑡)+^𝑡₀𝑓(𝑢)𝑦_𝑚−1(𝑢)𝑑𝑢+^𝑡₀¹₀𝑘₁(𝑢,𝑠)𝑦_𝑚−1(𝑠)𝑑𝑠𝑑𝑢+^𝑡₀^𝑢₀𝑘₂(𝑢,𝑠)𝑦_𝑚−1(𝑠)𝑑𝑠𝑑𝑢. (3.5)

The sequence {𝑦_𝑚}_𝑚≥0 converges to the solution 𝑥 of (1.1) and 𝑥−𝑦_𝑚≤𝑔𝛾^𝑚+1.1−𝛾 (3.6)

We can then calculate iteratively using (3.5), at least in a theoretical way, the solution of (1.1). From a practical viewpoint, in general these calculations are not possible explicitly. The idea of our numerical method is to use an appropriate Schauder basis in the spaces 𝒞([0,1]), 𝒞([0,1]²) truncating the functions of such spaces by means of the projections of the Schauder bases {𝑃_𝑛}_𝑛≥1 and {𝐵_𝑛}_𝑛≥1, and replacing each 𝐿𝑦_𝑚−1 (𝑚≥1) in (3.5) by a new function 𝑣_𝑚∈𝒞([0,1]), easier to calculate, and in such a way that the error ‖𝑥−(𝑔+𝑣_𝑚)‖ be small enough. Given these functions, each 𝑔+𝑣_𝑚 will be approximation of the solution of (1.1).

Specifically, we begin with 𝑣₀(𝑡)=0 and consider 𝑦₀(𝑡)=𝑦₀(𝑡)=𝑔(𝑡)+𝑣₀(𝑡). (3.7) Letting 𝑚∈ℕ, we inductively define the functions 𝑣_𝑚(𝑡)=^𝑡₀𝑃_𝑛𝑚𝑓(𝑢)𝑦_𝑚−1(𝑢)𝑑𝑢+^𝑡₀¹₀𝑄_𝑛2𝑚𝑘₁(𝑢,𝑠)𝑦_𝑚−1(𝑠)+𝑑𝑠𝑑𝑢_𝑡0𝑢0𝑄_𝑛2𝑚𝑘₂(𝑢,𝑠)𝑦_𝑚−1(𝑠)𝑑𝑠𝑑𝑢,𝑦_𝑚(𝑡)=𝑔(𝑡)+𝑣_𝑚(𝑡), (3.8) where 𝑛_𝑚 are natural numbers.

We will show that the sequence {𝑦_𝑚}_𝑚≥0 approximates the solution of (1.1) while in order to study the error ‖𝑥−𝑦_𝑚‖, let us assume that 𝑘₁,𝑘₂∈𝐶¹([0,1]²) and 𝑓,𝑔∈𝐶¹([0,1]).

In the first place, we show the following.

Lemma 3.1. The sequences {‖𝑣_𝑚‖}_𝑚∈ℕ and {‖𝑣^′_𝑚‖}_𝑚∈ℕ are bounded.

Proof. First we show, using an inductive argument, that for all 𝑡∈[0,1], |𝑣_𝑚(𝑡)|≤‖𝑔‖∑^𝑚_𝑗=1𝛾^𝑗. Since the Schauder basis {𝐵_𝑛}_𝑛≥1 is monotone, we have 𝑣₁(𝑡)≤𝑔𝑓^𝑡₀𝑑𝑢+𝑔𝑘₁^𝑡₀¹₀𝑑𝑠𝑑𝑢+𝑔𝑘₂^𝑡₀^𝑢₀𝑑𝑠𝑑𝑢≤𝑔𝛾. (3.9)
Suppose that the result holds for 𝑚−1, in other words |𝑣_𝑚−1(𝑡)|≤‖𝑔‖∑^𝑚−1_𝑗=1𝛾^𝑗, and using the monotony of {𝐵_𝑛}_𝑛≥1, we prove for 𝑚 the following: 𝑣_𝑚(𝑡)≤𝑓𝑦_𝑚−1^𝑡₀𝑑𝑢+𝑘₁𝑦_𝑚−1^𝑡₀¹₀+𝑘𝑑𝑠𝑑𝑢₂𝑦_𝑚−1^𝑡₀^𝑢₀𝑑𝑠𝑑𝑢≤𝑦_𝑚−1𝛾≤𝑔+𝑣_𝑚−1𝛾≤𝑔𝛾+𝑔^𝑚−1_𝑗=1𝛾^𝑗𝛾=𝑔^𝑚_𝑗=1𝛾^𝑗. (3.10) Therefore, ‖𝑣_𝑚‖≤‖𝑔‖∑_𝑗≥1𝛾^𝑗.
On the other hand, with similar arguments, 𝑣^′_𝑚(𝑡)≤𝑓𝑦_𝑚−1+𝑘₁𝑦_𝑚−1¹₀+𝑘𝑑𝑠₂𝑦_𝑚−1^𝑡₀𝑑𝑠≤𝑦_𝑚−1𝑓+𝑘₁+𝑘₂≤𝑦_𝑚−12𝛾≤2𝛾𝑔+𝑣_𝑚−1, (3.11) and thus, ‖𝑣^′_𝑚‖≤2𝛾(‖𝑔‖+‖𝑣_𝑚−1‖).

Remark 3.2. Given that for 𝑖∈{1,2} and 𝑚∈ℕ, 𝜕𝑘_𝑖(𝑡,𝑠)𝑦_𝑚(𝑠)=𝜕𝑡𝜕𝑘_𝑖(𝑡,𝑠)𝜕𝑡𝑦_𝑚(𝑠),𝜕𝑘_𝑖(𝑡,𝑠)𝑦_𝑚(𝑠)=𝜕𝑠𝜕𝑘_𝑖(𝑡,𝑠)𝜕𝑠𝑦_𝑚(𝑠)+𝑘_𝑖(𝑡,𝑠)𝑦^′_𝑚(𝑠), (3.12) denoting by 𝑀_𝑚=max𝑓′𝑦_𝑚+𝑓𝑦^′_𝑚,𝜕𝑘_𝑖𝑦_𝑚,𝜕𝑡𝜕𝑘_𝑖𝑦_𝑚𝜕𝑠_𝑖∈{1,2}, (3.13) we have that the sequence {𝑀_𝑚}_𝑚∈ℕ is also bounded.

Theorem 3.3. With the previous notation, if 𝑚∈ℕ, ℎ=max_{𝑗=0,…,𝑚−1}Δ𝑇_𝑛𝑗+1 with 𝑛_𝑗+1≥2 and 𝑀=sup{𝑀_𝑚}_𝑚∈ℕ, then 𝑥−𝑦_𝑚≤𝑔𝛾^𝑚+11−𝛾+8𝑀ℎ^𝑚−1_𝑗=0𝛾^{𝑚−1−𝑗}. (3.14)

Proof. The triangle inequality gives ‖𝑥−𝑦_𝑚‖≤‖𝑥−𝑦_𝑚‖+‖𝑦_𝑚−𝑦_𝑚‖. Because of the inequality (3.6), we have 𝑥−𝑦_𝑚≤𝑔𝛾^𝑚+1.1−𝛾 (3.15) For the second one, by an inductive argument, we can show that 𝑦_𝑚−𝑦_𝑚≤^𝑚−1_𝑗=0𝛾^{𝑚−1−𝑗}𝐿𝑦_𝑗−𝑣_𝑗+1. (3.16) Indeed, for 𝑚=1, the result is clearly true 𝑦₁−𝑦₁=𝑔+𝐿𝑔−𝑔−𝑣₁=𝐿𝑔−𝑣₁=𝐿𝑦₀−𝑣₁. (3.17) Suppose it holds for 𝑚−1, that is, ‖𝑦_𝑚−1−𝑦_𝑚−1‖≤∑^𝑚−2_𝑗=0𝛾^{(𝑚−1)−1−𝑗}‖𝐿𝑦_𝑗−𝑣_𝑗+1‖, and we prove for 𝑚 the following: 𝑦_𝑚−𝑦_𝑚=𝑔+𝐿𝑦_𝑚−1−𝑦_𝑚=𝐿𝑦_𝑚−1−𝑣_𝑚≤𝐿𝑦_𝑚−1−𝐿𝑦_𝑚−1+𝐿𝑦_𝑚−1−𝑣_𝑚≤‖𝐿‖𝑦_𝑚−1−𝑦_𝑚−1+𝐿𝑦_𝑚−1−𝑣_𝑚≤‖𝐿‖^𝑚−2_𝑗=0𝛾^{(𝑚−1)−1−𝑗}𝐿𝑦_𝑗−𝑣_𝑗+1+𝐿𝑦_𝑚−1−𝑣_𝑚≤𝛾^𝑚−2_𝑗=0𝛾^{(𝑚−1)−1−𝑗}𝐿𝑦_𝑗−𝑣_𝑗+1+𝐿𝑦_𝑚−1−𝑣_𝑚=^𝑚−1_𝑗=0𝛾^{𝑚−1−𝑗}𝐿𝑦_𝑗−𝑣_𝑗+1. (3.18) Then 𝑥−𝑦_𝑚≤𝑔𝛾^𝑚+1+1−𝛾^𝑚−1_𝑗=0𝛾^{𝑚−1−𝑗}𝐿𝑦_𝑗−𝑣_𝑗+1. (3.19) For 𝑗=0,1,…,𝑚−1, 𝐿𝑦_𝑗−𝑣_𝑗+1(𝑡)=^𝑡₀𝑓(𝑢)𝑦_𝑗(𝑢)𝑑𝑢+^𝑡₀¹₀𝑘₁(𝑢,𝑠)𝑦_𝑗(𝑠)𝑑𝑠𝑑𝑢+^𝑡₀^𝑢₀𝑘₂(𝑢,𝑠)𝑦_𝑗(𝑠)𝑑𝑠𝑑𝑢−^𝑡₀𝑃_𝑛𝑗+1𝑓(𝑢)𝑦_𝑗(𝑢)𝑑𝑢−^𝑡₀¹₀𝑄_𝑛²_𝑗+1𝑘₁(𝑢,𝑠)𝑦_𝑗(𝑠)𝑑𝑠𝑑𝑢−^𝑡₀^𝑢₀𝑄_𝑛²_𝑗+1𝑘₂(𝑢,𝑠)𝑦_𝑗(𝑠)≤𝑑𝑠𝑑𝑢_𝑡0𝑓(𝑢)𝑦_𝑗(𝑢)−𝑃_𝑛𝑗+1𝑓(𝑢)𝑦_𝑗(𝑢)+𝑑𝑢_𝑡010𝑘₁(𝑢,𝑠)𝑦_𝑗(𝑠)−𝑄_𝑛²_𝑗+1𝑘₁(𝑢,𝑠)𝑦_𝑗(𝑠)𝑑𝑠𝑑𝑢+^𝑡₀^𝑢₀𝑘₂(𝑢,𝑠)𝑦_𝑗(𝑠)−𝑄_𝑛²_𝑗+1𝑘₂(𝑢,𝑠)𝑦_𝑗(𝑠)≤𝑑𝑠𝑑𝑢𝑓𝑦_𝑗−𝑃_𝑛𝑗+1𝑓𝑦_𝑗𝑡+𝑘₁𝑦_𝑗−𝑄_𝑛²_𝑗+1𝑘₁𝑦_𝑗𝑡+𝑘₂𝑦_𝑗−𝑄_𝑛²_𝑗+1𝑘₂𝑦_𝑗𝑡²2≤𝑓𝑦_𝑗−𝑃_𝑛𝑗+1𝑓𝑦_𝑗+𝑘₁𝑦_𝑗−𝑄_𝑛²_𝑗+1𝑘₁𝑦_𝑗+𝑘₂𝑦_𝑗−𝑄_𝑛²_𝑗+1𝑘₂𝑦_𝑗12. (3.20)
And thus, applying (2.7), we obtain the following bound: 𝐿𝑦_𝑗−𝑣_𝑗+1≤2𝑀_𝑗ℎ+4𝑀_𝑗ℎ+4𝑀_𝑗ℎ12=8𝑀_𝑗ℎ≤8𝑀ℎ. (3.21)
The proof is complete in view of the triangular inequality and of (3.6) and (3.16).

Remark 3.4 3.4. Under the hypothesis of Theorem 3.3, for all 𝜀>0, we can find 𝑚∈ℕ and positive integers 𝑛₁,…,𝑛_𝑚 such that ‖𝑥−𝑦_𝑚‖<𝜀. Furthermore, since 𝛾<1 and the points of partition can be chosen such that ℎ is as small as we desire, ‖𝑥−𝑦_𝑚‖ is as small as we desire, and (3.14) also provides a quota of the error committed when we approach 𝑥 by 𝑦_𝑚.

Remark 3.5. It follows from Theorem 3.3 that the proposed method with the chosen Schauder bases has order of convergence one. This choice has been done by simplicity in the exhibition of the results, and it has allowed us to obtain satisfactory numerical results as we show in the following section. Nevertheless, by integrating the considered basis in 𝒞([0,1]) (and adding the one constant function), we obtain new bases of 𝒞([0,1]) and 𝒞([0,1]²). Considering these bases, the order of convergence is 2. In general, integrating 𝑛+1 times, we would obtain order of convergence 𝑛.

4. Some Numerical Example

We now turn our attention to show two numerical performance results. For each example, we have fixed the subset {𝑡_𝑖}_𝑖≥1 chosen for constructing the Schauder basis {𝑏_𝑛}_𝑛≥1 in 𝒞([0,1]) and {𝐵_𝑛}_𝑛≥1 in 𝒞([0,1]²), specifically, 𝑡₁=0, 𝑡₂=1; and for 𝑛∈ℕ∪{0}, 𝑡_𝑖+1=(2𝑘+1)/2^𝑛+1 if 𝑖=2^𝑛+𝑘+1 where 0≤𝑘<2^𝑛 are integers. To define the sequence {𝑦_𝑚}, we take 𝑛_𝑗=𝑖 (for all 𝑗≥1). In addition we include, a table exhibiting, for 𝑖=9,17, and 33, the absolute errors committed for certain representative points of [0,1] when we approximate the exact solution 𝑥(𝑡) by means of the iteration 𝑦_𝑚(𝑡) where 𝑚 is shown in the table. The algorithms associated with the numerical method were performed using Mathematica 7. We have checked that when more points are used, the accuracy improves significantly, unlike the number of iterations.

In order to provide some details, we synthesize the steps of the programmed algorithm: (1)we introduce the nodes; we construct the base {𝑏_𝑛} and the base {𝐵_𝑛}, (2)we calculate 𝑔 (if it is not possible to explicitly arrive at 𝑔, we can apply a quadrature method), (3)we define 𝑣₀(𝑡)=0 and 𝑦₀(𝑡)=𝑔(𝑡)+𝑣₀(𝑡), (4)we calculate 𝑓𝑦₀ and 𝑃_𝑛1(𝑓𝑦₀) using the base {𝑏_𝑛} and 𝑘₁𝑦₀, 𝑘₂𝑦₀, 𝑄_𝑛²₁(𝑘₁𝑦₀), and 𝑄_𝑛²₁(𝑘₂𝑦₀) using the base {𝐵_𝑛}. Note that we calculate the projections by integrals of piecewise univariate and bivariate polynomials of degrees 1 and 2 and the calculation of the coefficients of such polynomials just requires linear combinations of several evaluations of the basic functions at adequate points. We do not have to solve systems of algebraic equations, (5)we calculate the expression 𝑦₁ using Definition (3.8), (6)repeat the process.

Example 4.1. For the first example, we consider the following linear Fredholm-Volterra integro-differential equation with the exact solution 𝑥(𝑡)=𝑡². Its numerical results are given in Table 1 and Figure 1: 𝑥^′(𝑡)=112010+180𝑡−15𝑡³−6𝑡⁵1𝑡+8(𝑡)+𝑡𝑥₁₀(2−3(𝑠+𝑡)+6𝑠𝑡)𝑥(𝑠)𝑑𝑠+¹₀𝑠𝑡5𝑥(𝑠)𝑑𝑠(𝑡,𝑠)∈[0,1]²,𝑥(0)=0. (4.1)

Table 1 

Absolute errors for Example 4.1.

Figure 1 

The plot of absolute errors by our method for Example 4.1.

Example 4.2. For the second example, we consider the following linear Fredholm-Volterra integro-differential equation with the exact solution 𝑥(𝑡)=𝑡. Its numerical results are given in Table 2 and Figure 2: 𝑥^′(𝑡)1=1−−14𝜋8𝑡²(𝑡)−1cos3(𝑡)+(𝑡)𝑡sinsin3𝑥(𝑡)+14¹₀()𝑥(𝑠)1sin𝜋𝑠𝑑𝑠+4¹₀cos(𝑡)𝑥(𝑠)𝑑𝑠(𝑡,𝑠)∈[0,1]²,𝑥(0)=0. (4.2)

Table 2 

Absolute errors for Example 4.2.

Figure 2 

The plot of absolute errors by our method for Example 4.2.

5. Conclusions

An efficient approach easy to implement is proposed to solve the linear Fredholm-Volterra integro-differential equations. The approximating functions are the sum of integrals of piecewise bivariate polynomials of degree 2, and the calculation of the coefficients of such polynomials just requires linear combinations of several evaluations of the basic functions at adequate points. The approach leads to an approximate solution of the integro-differential equation, which can be expressed explicitly in simple closed form, and which can be effectively computed using symbolic computing codes on any personal computer.

Acknowledgments

This research is partially supported by Junta de Andalucía Grant FQM359 and the ETSIE of the University of Granada.