Numerical Solutions of a Class of Nonlinear Volterra Integral Equations

Abstract

We consider numerical solutions of a class of nonlinear (nonstandard) Volterra integral equations. We first prove the existence and uniqueness of the solution of the Volterra integral equation in the context of the space of continuous functions over a closed interval. We then use one-point collocation methods with a uniform mesh to construct solutions of the nonlinear (nonstandard) VIE and quadrature rules. We conclude that the repeated Simpson's rule gives better solutions when a reasonably large value of the stepsize is used.

1. Introduction

In this paper we study the nonlinear (nonstandard) Volterra integral equation of the second kind of the form 𝑢(𝑡)=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑢(𝑠)𝑑𝑠^𝑗,𝑡∈[0,𝑇], (1) where (𝑟∈ℕ, 𝑟≥2), with 𝑏_𝑗∈ℝ, and 𝑔_𝑗, 𝑘_𝑗 are continuous functions. Volterra integral equations play an important part in scientific and engineering problems such as population dynamics, spread of epidemics, semiconductor devices, wave propagation, superfluidity, and travelling wave analysis, Saveljeva [1]. In cases where the kernel is of convolution type (𝐾(𝑡,𝑠)=𝐾(𝑡−𝑠)) the solutions to (1) include elliptic functions and natural generalizations of these functions which also have wide applications in the fields of science and engineering [2]. This class of Volterra integral equations was considered by Sloss and Blyth [2] who proved the existence and uniqueness of the solution in the Banach space 𝐿² and applied the Corrington’s Walsh function method to (1).

Much work has been done in the study of numerical solutions to Volterra integral equations using collocation methods [1, 3–7]. Benitez and Bolos [8] pointed out that collocation methods have proven to be a very suitable technique for approximating solutions to nonlinear integral equations because of their stability and accuracy. Other authors such as [9–12] used quadrature rules like repeated trapezoidal and repeated Simpson’s rule to solve linear Volterra integral equations. However, collocation methods and quadrature rules have not been used to approximate solutions to (1).

2. The Numerical Methods

2.1. The Collocation Method

In our work we focus on one-point collocation methods (see [13]).

Let 𝑡_𝑛∶=𝑛ℎ(𝑛=0,1,…,𝑁−1) define a uniform partition for 𝐼=[0,𝑇] and set 𝑍_𝑁∶=𝑡₀,…,𝑡_𝑁,𝐼₀∶=[𝑡₀,𝑡₁]𝐼_𝑛∶=(𝑡_𝑛,𝑡_𝑛+1](1≤𝑛≤𝑁−1). The solution to (1) will be approximated by using collocation in the piecewise constant polynomial space 𝑆₀⁻¹(𝑍_𝑁).

For a given real number 𝑐₁, define the set 𝑋_𝑁∶=𝑡_𝑛,1 of collocation points by 𝑡_𝑛,1=𝑡_𝑛+𝑐₁ℎ0≤𝑐₁≤1,𝑛=0,…,𝑁−1. (2)

The collocation solution 𝑢_𝑛∈𝑆₀⁻¹(𝑍_𝑁) is defined by the collocation equation 𝑢_𝑛(𝑡)=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑢(𝑠)𝑑𝑠^𝑗,𝑡∈𝑋_𝑁, (3) since 𝑢_𝑛(𝑡)=𝑢_𝑛𝑡_𝑛+𝜈ℎ=𝐿₁(𝜈)𝑈_𝑛,1,𝜈∈(0,1], (4) where 𝐿₁(𝜈)=1 and is a Lagrange fundamental polynomial.

Thus for 𝑡=𝑡_𝑛,1∶=𝑡_𝑛+𝑐₁ℎ and 0<𝑐₁≤1 the collocation equation (3) assumes the form 𝑢_𝑛(𝑡)=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡)+^𝑡𝑛₀𝑘_𝑗(𝑡,𝑠)𝑢_𝑖(𝑠)+ℎ^𝑐1₀𝑘_𝑗𝑡,𝑡_𝑛+𝑠ℎ𝑢_𝑛𝑡_𝑛+𝑠ℎ𝑑𝑠^𝑗. (5)

Expressing the collocation equation in terms of the stage values we get 𝑈_𝑛,1=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡_𝑛,1+𝐹_𝑗𝑛𝑡_𝑛,1+ℎ^𝑐1₀𝑘_𝑗(𝑡_𝑛,1,𝑡_𝑛+𝑠ℎ)𝑑𝑠𝑈_𝑛,1^𝑗. (6) Let 𝑡∈𝐼_𝑛 and define 𝐹_𝑗𝑛(𝑡)∶=^𝑡𝑛₀𝑘_𝑗(𝑡,𝑠)𝑢_𝑖(𝑠)𝑑𝑠. (7) Then 𝐹_𝑗𝑛𝑡_𝑛,1=ℎ^𝑛−1_𝑖=0¹₀𝑘_𝑗𝑡_𝑛,1,𝑡_𝑖+𝑠ℎ𝑑𝑠𝑈_𝑖,1. (8) The term 𝐹_𝑗𝑛(𝑡_𝑛,1) is called the lag term corresponding to the collocation solution, [13].

Iterated Collocation. The iterated approximation 𝑢^𝐼 corresponding to 𝑢 is defined by 𝑢^𝐼(𝑡)=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑢(𝑠)𝑑𝑠^𝑗𝑡∈𝐼 (9) (see [4, 5, 14]).

Set 𝑡=𝑡_𝑛∈𝑍_𝑁 and use (4); we may write (9) in the form 𝑢^𝐼𝑡_𝑛=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡_𝑛)+ℎ^𝑛−1_𝑖−0¹₀𝑘_𝑗(𝑡_𝑛,𝑡_𝑖+𝑠ℎ)𝑑𝑠𝑈_𝑖,1^𝑗. (10)

2.2. Repeated Trapezoidal Rule

Using the trapezoidal rule we construct the solution to the integral equation (1) (see [12]). Let 𝑡₀=𝑎,𝑡_𝑛𝑡=𝑏_𝑖=𝑡₀𝑢𝑡+𝑖ℎ𝑖=0,1,2,…,𝑛_𝑖=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡_𝑖+^𝑖_𝑙=1^𝑡𝑖_𝑡𝑙−1𝑘_𝑗𝑡_𝑖,𝑠𝑢(𝑠)𝑑𝑠^𝑗𝑖=1,2,…,𝑛. (11)

The approximation of the integral in (11) by repeated trapezoidal rule will give the following system: 𝑢𝑡₀=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗^𝑗,𝑢𝑡₁=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡₁+ℎ2𝑘_𝑗(𝑡₁,𝑡₀)𝑢(𝑡₀)+𝑘_𝑗(𝑡₁,𝑡₁)𝑢(𝑡₁)^𝑗,𝑢𝑡₂=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡₂+ℎ2𝑘_𝑗𝑡₂,𝑡₀𝑢𝑡₀+ℎ𝑘_𝑗𝑡₂,𝑡₁𝑢𝑡₁+ℎ2𝑘_𝑗𝑡₂,𝑡₂𝑢𝑡₂^𝑗,𝑢₃=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡₃+ℎ2𝑘_𝑗𝑡₃,𝑡₀𝑢𝑡₀+ℎ𝑘_𝑗𝑡₃,𝑡₁𝑢𝑡₁+𝑘_𝑗𝑡₃,𝑡₂𝑢𝑡₂+ℎ2𝑘_𝑗𝑡₃,𝑡₃𝑢𝑡₃^𝑗,⋮𝑢𝑡_𝑛=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡_𝑛+ℎ2𝑘_𝑗𝑡_𝑛,𝑡₀𝑢𝑡₀+ℎ𝑘_𝑗𝑡_𝑛,𝑡₁𝑢𝑡₁+⋯+ℎ𝑘_𝑗𝑡_𝑛,𝑡_𝑛−1×𝑢(𝑡_𝑛−1ℎ)+2𝑘_𝑗(𝑡_𝑛,𝑡_𝑛)𝑢(𝑡_𝑛)^𝑗. (12)

2.3. Repeated Simpson’s Rule

We use repeated Simpson’s rule to construct the solution to the integral equation (1) (see [9]).

If 𝑛 is even, then Simpson’s rule may be applied to each subinterval [𝑡_2𝑖,𝑡_2𝑖+1,𝑡_2𝑖+2]. For 𝑖=0,1,…,(𝑁/2)−1 we have ^𝑡2𝑖+2_𝑡2𝑖𝑓(𝑡)ℎ𝑑𝑠≃3𝑓𝑡_2𝑖+4𝑓𝑡_2𝑖+1+𝑓𝑡_2𝑖+2. (13)

Summing up, ^𝑏_𝑎𝑓(𝑡)ℎ𝑑𝑡=3^𝑁−1_𝑙=0𝑓𝑡_2𝑖𝑡+4𝑓^2𝑙+𝑡_2𝑙+22+𝑓𝑡_2𝑙+2. (14) We use (14) to solve the nonlinear (nonstandard) VIE. The approximation of (1) in the even nodes 𝑡_2𝑚 is given by 𝑢_2𝑚=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡_2𝑚)+^𝑡2𝑚_𝑎𝑘_𝑗(𝑡_2𝑚,𝑠)𝑢(𝑠)𝑑𝑠^𝑗. (15)

Using 𝑢𝑡_2𝑙+1≃𝑢𝑡_2𝑙+𝑢𝑡_2𝑙+22. (16)

we obtain 𝑢𝑡_2𝑚=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡_2𝑚+ℎ3^𝑚−1_𝑙=0𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙𝑢𝑡_2𝑙+4𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙+1𝑢𝑡_2𝑙+𝑢𝑡_2𝑙+22+𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙+2𝑢(𝑡_2𝑙+2)^𝑗𝑢𝑡_2𝑚=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡_2𝑚+ℎ3^𝑚−1_𝑙=0𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙+2𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙+1𝑢𝑡_2𝑙+𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙+1+2𝑘_𝑗𝑡_2𝑚,𝑡_𝑙+1×𝑢𝑡_2𝑙+2^𝑗𝑢𝑡_2𝑚=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗𝑡_2𝑚+ℎ3𝑘_𝑗𝑡_2𝑚,𝑡₀+2𝑘_𝑗𝑡_2𝑚,𝑡₁+ℎ3𝑘_𝑗𝑡_2𝑚,𝑡_2𝑚+𝑘_𝑗𝑡_2𝑚,𝑡_2𝑚−1𝑢𝑡_2𝑚+2ℎ3^𝑚−1_𝑙=0𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙−1+𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙+𝑘_𝑗𝑡_2𝑚,𝑡_2𝑙+1𝑢(𝑡_2𝑙)^𝑗. (17)

3. Existence and Uniqueness of the Solution

The following theorem shows that when 𝑟=2 and 𝑏₁=0 the integral equation (1) has a unique solution in the space 𝐶[0,𝑑]. Theorem 2 gives sufficient conditions for the solution to (1) to exist. We prove the existence and uniqueness of the solution using a procedure analogous to the one used in Sloss and Blyth [2].

Theorem 1. The integral equation 𝑧(𝑡)=𝑏𝑔(𝑡)+^𝑡₀𝑘(𝑡,𝑠)𝑧(𝑠)𝑑𝑠², (18) with 𝑔∈𝐶[0,1], 𝑏∈ℝ, and 𝑘(𝑡,𝑠)∈𝐶([0,1]×[0,1]), has a unique solution 𝑢 and the solution belongs to 𝐼_𝑑=[0,𝑑], 0<𝑑≤1, with 10<𝑑<𝐾1|𝑏|−𝑔2𝐾^∞−‖𝑢‖_∞, (19) where 𝐾=sup[0,1]×[0,1]|𝑘(𝑡,𝑠)|. (20)

Proof. The existence of the solution is shown in the corollary of Theorem 2 (in the next section). Here we prove the uniqueness of the solution. Let 𝑢 and 𝑢+𝑣 be solutions of (18).
Then, 𝑣(𝑡)=𝑏𝑔(𝑡)^𝑡₀𝑘(𝑡,𝑠)𝑣(𝑠)𝑑𝑠+2^𝑡₀𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠⋅^𝑡₀𝑘()𝑣(𝑠)+𝑡,𝑠_𝑡0𝑘(𝑡,𝑠)𝑣(𝑠)𝑑𝑠². (21)
Define 𝑇_𝑛∶𝐶[0,1]×𝐶[0,1] by 𝑇_𝑛𝑧(𝑡)=𝜒_𝑛𝑏2𝑔(𝑡)^𝑡₀𝑘(𝑡,𝑠)𝑧(𝑠)𝑑𝑠+2^𝑡₀𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠⋅^𝑡₀𝑘()𝑧(𝑠)+𝑡,𝑠𝑑𝑠_𝑡0𝑘(𝑡,𝑠)𝑧(𝑠)𝑑𝑠², (22) where 𝜒_𝑛 is a sequence of characteristic functions of intervals [0,𝐴_𝑛]⊂[0,1].
Let 𝑣₁+𝑣₂∈𝐶[0,1]; consider 𝑇_𝑛𝑣₁(𝑡)−𝑇_𝑛𝑣₂(𝑡)=𝜒_𝑛(𝑡)𝑏2𝑔(𝑡)+2^𝑡₀𝑘()𝑢(𝑠)+𝑡,𝑠𝑑𝑠_𝑡0𝑘(𝑡,𝑠)𝑣₁(𝑠)+𝑣₂(𝑠)𝑑𝑠⋅^𝑡₀𝑘(𝑡,𝑠)𝑣₁(𝑠)−𝑣₂(𝑠)𝑑𝑠_def=𝑏𝑧𝑢,𝑣₁,𝑣₂(𝑡)^𝑡₀𝑘(𝑡,𝑠)𝑣₁(𝑠)−𝑣₂(𝑠)𝑑𝑠; (23) then, 𝑇_𝑛𝑣₁(𝑡)−𝑇_𝑛𝑣₂(𝑡)_∞≤𝑏𝑧¹₀𝐾(𝑡,𝑠)𝑣₁(𝑠)−𝑣₂(𝑠)𝑑𝑠_∞. (24)
Let 𝑑=max_0≤𝑡≤1(|𝑣₁(𝑡)|,|𝑣₂(𝑡)|) and take 𝑇_𝑛∶[0,𝑑]→[0,1].
Then, ‖𝑧‖_∞≤2𝑔_∞+2𝐾‖𝑢‖_∞+2𝐾𝑑; (25) therefore 𝑇_𝑛𝑣₁(𝑡)−𝑇_𝑛𝑣₂(𝑡)_∞≤|𝑏|2𝑔_∞+2𝐾‖𝑢‖_∞+2𝐾𝑑𝐾𝑣₁−𝑣_2∞. (26) Thus 𝑇_𝑛 is contractive if 2|𝑏|𝐾𝑔_∞+2𝐾²‖𝑢‖_∞|𝑏|<1. (27) That is, 10<𝑑<𝐾1|𝑏|−𝑔2𝐾^∞−‖𝑢‖_∞. (28)
Clearly, 𝑇_𝑛 maps 𝐶[0,𝑑] onto itself if (19) is satisfied. Also, 𝑇_𝑛(0)=0.
Suppose 𝑣(𝑡)≠0 is a solution of (21), such that 𝑣 may lie outside of [0,𝑑]. Then, 𝜒_𝑛𝑣(𝑡)=𝜒_𝑛𝑏2𝑔(𝑡)^𝑡₀𝑘(𝑡,𝑠)𝑣(𝑠)𝑑𝑠+2^𝑡₀𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠⋅^𝑡₀𝑘()𝑣(𝑠)+𝑡,𝑠𝑑𝑠_𝑡0𝑘(𝑡,𝑠)𝑣(𝑠)𝑑𝑠²=𝑏2𝑔(𝑡)𝜒_𝑛(𝑡)^𝑡₀𝑘(𝑡,𝑠)𝜒_𝑛(𝑠)𝑣(𝑠)𝑑𝑠+2^𝑡₀𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠𝜒_𝑛(𝑡)×^𝑡₀𝑘(𝑡,𝑠)𝜒_𝑛(𝑠)𝑣(𝑠)+𝜒𝑑𝑠_𝑛(𝑡)^𝑡₀𝑘(𝑡,𝑠)𝜒_𝑛(𝑠)𝑣(𝑠)𝑑𝑠²,=𝑇_𝑛𝜒_𝑛𝑣(𝑡), (29) which shows that 𝜒_𝑛𝑣 is a fixed point of 𝑇_𝑛 for all 𝑛. Since 𝜒_𝑛𝑦→0 in 𝐶[0,1] as 𝐴_𝑛→0, and for 𝑣≠0, we can select 𝜒_𝑛𝑣∈[0,𝑑]. But this is impossible since 0 is the only solution in [0,𝑑]. Therefore the solution 𝑢 of (18) is unique in 𝐶[0,𝑑] if 𝑑>0 exists that satisfies (19).

Theorem 2. There exists a solution 𝑢 of (1), where 𝑢∈𝐶[0,𝑑] provided that 𝑁_𝑏^𝑟_𝑗=1𝑗𝑏_𝑗𝑔_𝑗∞+𝐾_𝑗𝑑^𝑗−1𝐾_𝑗<1,_𝑟𝑗=1𝑏_𝑗𝑔_𝑗∞+𝐾_𝑗𝑑<𝑑, (30) where 𝑁_𝑏 is the number of nonzero 𝑏_𝑗.

Proof. Let 𝑇^′𝑣(𝑡)=∑^𝑟_𝑗=1𝑏_𝑗(𝑔_𝑗(𝑡)+∫^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣(𝑠)𝑑𝑠)^𝑗 and 𝑣₁,𝑣₂∈[0,𝑑] for a suitable 𝑑, and consider 𝑇^′𝑣₂(𝑡)−𝑇^′𝑣₁(𝑡)=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣₂(𝑠)𝑑𝑠^𝑗−𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣₁(𝑠)𝑑𝑠^𝑗=^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣₂(𝑠)𝑑𝑠−𝑔_𝑗(𝑡)−^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣₁(𝑠)⋅𝑑𝑠_{𝑗−1𝑖=0}𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣₂(𝑠)^𝑗⋅𝑔𝑑𝑠_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣₁(𝑠)^𝑗𝑑𝑠^{𝑗−1−𝑖}_def=^𝑟_𝑗=1𝑏_𝑗𝐹_𝑗𝑡,𝑣₁,𝑣₂^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣₂(𝑠)−𝑣₁(𝑠)𝑑𝑠. (31) So 𝑇^′𝑣₂−𝑇^′𝑣_1∞≤𝑁_𝑏^𝑟_𝑗=1𝑏_𝑗sup𝐹_𝑗𝑡,𝑣₁,𝑣₂⋅𝐾_𝑗𝑣₂−𝑣_1∞;𝐹_𝑗∞≤𝑗𝑔_𝑗∞+𝐾_𝑗𝑑^𝑗−1 (32) therefore 𝑇^′𝑣₂−𝑇^′𝑣_1∞≤𝑁_𝑏^𝑟_𝑗=1𝑗𝑏_𝑗𝑔_𝑗∞+𝐾_𝑗𝑑^𝑗−1𝐾_𝑗𝑣₂−𝑣_1∞. (33)
Consequently 𝑇^′ is a contraction mapping if 𝑁_𝑏^𝑟_𝑗=1𝑗𝑏_𝑗𝑔_𝑗∞+𝐾_𝑗𝑑^𝑗−1𝐾_𝑗<1. (34)
We need to show that 𝑇^′∶𝐶[0,𝑑]→𝐶[0,𝑑]. Observe that 𝑔_𝑗(𝑡)+^𝑡₀𝑘_𝑗(𝑡,𝑠)𝑣(𝑠)𝑑𝑠^𝑗_∞≤𝑔_𝑗∞+𝐾_𝑗‖𝑣‖_∞^𝑗. (35)
Therefore 𝑇^′𝑣_∞≤^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗∞+𝐾_𝑗‖𝑣‖_∞^𝑗≤^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗∞+𝐾_𝑗𝑑^𝑗; (36)
thus 𝑇^′∶𝐶[0,𝑑]→𝐶[0,𝑑] if ^𝑟_𝑗=1𝑏_𝑗𝑔_𝑗∞+𝐾_𝑗𝑑^𝑗<𝑑. (37) Hence the map 𝑇^′ is a contraction and maps [0,𝑑] into itself provided (30) is satisfied.

Corollary 3. There exists a solution 𝑢 to the integral equation 𝑢(𝑡)=𝑏𝑔(𝑡)+^𝑡₀𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠², (38) where 𝑢∈𝐶[0,𝑑] with 10<𝑑<𝐾12|𝑏|𝐾−𝑔_∞, (39) if 𝑔_∞<1|𝑏|.4𝐾 (40)

Proof. From Theorem 2 we get sufficient conditions for the existence of a solution 2𝐾|𝑏|𝑔_∞+𝐾𝑑<1,|𝑏|𝑔_∞+𝐾𝑑²<𝑑. (41)(42)
Inequality (41) is solved by any 𝑑<𝑑, where 1𝑑=𝐾12|𝑏|𝐾−𝑔_∞, (43) and inequality (42) is equivalent to |𝑏|𝐾²𝑑²+2𝐾|𝑏|𝑔_∞−1𝑑+|𝑏|𝑔²_∞<0. (44) This is satisfied by 𝑑∈(𝑑₋,𝑑₊), where 𝑑_±=1−2𝐾|𝑏|𝑔_∞±1−4𝐾|𝑏|𝑔_∞2|𝑏|𝐾². (45)
If the regularity condition 1−4𝐾|𝑏|𝑔_∞>0 (46) is satisfied, 𝑑₊ and 𝑑₋ are real and positive. Furthermore, 𝑑𝑑=⁻+𝑑₊2, (47) so that (46) ensures 𝑑∈(𝑑₋,𝑑) satisfies both inequalities (41) and (42) in Theorem 2.

4. Numerical Computations

In our work we consider examples of (1) when 𝑟=2. We use (6) to approximate the solutions considering two special cases: 𝑐₁=1/2 (implicit midpoint method) and 𝑐₁=1 (implicit Euler method). We also use the repeated trapezoidal and repeated Simpson’s rule. Since the methods are implicit we perform an iterative procedure at each step implementing a tolerance of 10⁻⁴. For each method we used three different values of ℎ: ℎ=0.01, ℎ=0.005, and ℎ=0.0025.

4.1. Example  1

Consider the nonlinear VIE 𝑢(𝑡)=21+^𝑡₀(𝑡−𝑠)𝑢(𝑠)𝑑𝑠²0≤𝑡≤1, (48) which arises from a nonlinear differential equation in [15] where 𝑏₁=0 and 𝑏₂=2.

4.1.1. Using Implicit Euler Method

When 𝑐₁=1 and 𝑡_𝑛,1=𝑡_𝑛+ℎ, the collocation solution of (48) is given by 𝑈_𝑛,1=21+𝐹_𝑛(𝑡_𝑛,1)+𝑈_𝑛,1ℎ²2², (49) where 𝐹_𝑛𝑡_𝑛,1=ℎ^𝑛−1_𝑖=0𝑡_𝑛−𝑡_𝑖+ℎ2𝑈_𝑖,1. (50)

Figure 1 shows the solution to (48) at ℎ=0.01, ℎ=0.005, and ℎ=0.0025.

Figure 1 

The collocation solution of (48) when 𝑐1=1.

4.1.2. Using Implicit Midpoint Method

When 𝑐₁=1/2 and 𝑡_𝑛,1=𝑡_𝑛+(ℎ/2), the collocation solution of (48) is given by 𝑈_𝑛,1=21+𝐹_𝑛(𝑡_𝑛,1)+𝑈_𝑛,1ℎ²8², (51) where 𝐹_𝑛𝑡_𝑛,1=ℎ^𝑛−1_𝑖=0𝑡_𝑛−𝑡_𝑖𝑈_𝑖,1. (52)

Figure 2 shows the solution to (48) at ℎ=0.01, ℎ=0.005, and ℎ=0.0025.

Figure 2 

The collocation solution of (48) when 𝑐1=1/2.

4.1.3. Using the Iterated Collocation

For 𝑐₁=1/2 the iterated collocation solution of (48) is given as 𝑢^𝐼𝑡_𝑛=21+ℎ^𝑛−1_𝑖=0¹₀𝑡_𝑛−𝑡_𝑖−𝑠ℎ𝑑𝑠𝑈_𝑖1². (53)

Integrate to obtain 𝑢^𝐼𝑡_𝑛=21+ℎ^𝑛−1_𝑖=0𝑡_𝑛−𝑡_𝑖−ℎ2𝑈_𝑖1². (54)

The iterated collocation solution of (48) with three different values of ℎ is shown in Figure 3.

Figure 3 

The iterated collocation solution of (48) when 𝑐1=1/2.

4.1.4. Using Repeated Trapezoidal Rule

For the VIE (48) 𝑢(0)=2 and 𝑢𝑡_𝑛ℎ=21+2𝑡_𝑛𝑢(0)+ℎ^𝑛−1_𝑖=1(𝑡_𝑛−𝑡_𝑛−1)𝑢_𝑛−1². (55)

Figure 4 shows the solution to the VIE (48) for the three values of ℎ used.

Figure 4 

The solution of (48) by the repeated trapezoidal rule.

4.1.5. Using Repeated Simpson’s Rule

When 𝑡=0, 𝑢(0)=2 for (48) and 𝑢𝑡_2𝑚ℎ=21+33𝑡_2𝑚−2𝑡₁𝑢(0)+2𝑡_2𝑚−2𝑡_2𝑚−1𝑢𝑡_2𝑚+2ℎ3^𝑚−1_𝑙=0(3𝑡_2𝑚−𝑡_2𝑙+1−𝑡_𝑙−𝑡_2𝑙−1)𝑢(𝑡_2𝑙)². (56)

The solution to (48) using repeated Simpson’s rule is shown in Figure 5.

Figure 5 

The solution of (48) by the repeated Simpson’s rule.

Table 1 shows the errors in the solution of the integral equation (48) for the largest value of ℎ used.

Table 1 

Absolute errors in the solution of (48) when ℎ=0.01.

4.2. Example  2

Consider 𝑢(𝑡)=1+^𝑡₀()𝑢(𝑠)+1𝑡−𝑠𝑑𝑠21+^𝑡₀(𝑡−𝑠)𝑢(𝑠)𝑑𝑠²0≤𝑡≤1, (57) where 𝑏₁=1 and 𝑏₂=1/2. The integral equation (57) arises from nonlinear differential equations that represent conservative systems (see [16]). We used the four methods to approximate the solution to this example and Example 3, and we present tables for the absolute errors in the solution. Table 2 shows the errors in the solution of (57) when ℎ=0.01:

Table 2 

Absolute errors in the solution of (57) when ℎ=0.01.

4.3. Example  3

Consider the integral equation 𝑢(𝑡)=21+^𝑡₀()𝑢(𝑠)+𝑡−𝑠𝑑𝑠1+^𝑡₀(𝑡−𝑠)𝑢(𝑠)𝑑𝑠²0≤𝑡≤1, (58) where 𝑏₁=2 and 𝑏₂=1. The nonlinear VIE arises from a nonlinear differential equation in [17]. Shown in Table 3 are the errors in the solution of (58) when ℎ=0.01.

Table 3 

Absolute errors in the solution of (58) when ℎ=0.01.

5. Discussion

We approximated the solutions to Examples  1–3 using the implicit Euler method, implicit midpoint method, and repeated trapezoidal and repeated Simpson’s rule using various values of the stepsize. At ℎ=0.001 and below we obtained a similar solution from all the methods used; hence we take that as our “exact” solution. Therefore, for sufficiently small ℎ we get a good accuracy of the numerical solutions. When the stepsize is greater than 0.001 we obtained different numerical solutions from each of the four methods. We use the “exact” solution and absolute error to study the performance of each method when the stepsize is increased.

Tables 1–3 show the absolute errors in the solutions when ℎ=0.01. From these tables we observe that the repeated Simpson’s rule performs better followed by the implicit midpoint method then the repeated trapezoidal rule. Among the four methods used, the implicit Euler method gives a larger error as h is increased. We then found an iterated collocation solution for the implicit midpoint method and the accuracy of the method improved as shown in Figure 3. According to our numerical results, we conclude that the repeated Simpson’s rule performs well since it gives better solutions when a reasonably large value of the stepsize is used. These observations are consistent for all three examples used.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.