Modified Optimization Algorithms for Infinite Integro-Differential Equations via Generalized Laguerre Polynomials

Integro-differential equations play a pivotal role in diverse scientific realms, spanning fluid dynamics [1, 2], biological models [3], and astrophysics [4, 5], and they are used to formulate a variety of physical phenomena. Their ubiquity stems from their ability to model a wide range of physical phenomena [6-9]. However, due to their inherent complexity, deriving analytical solutions for these equations proves challenging, necessitating the development of practical approximations, as it is difficult to solve them analytically. Recently, considerable attention has been directed towards using orthogonal polynomial techniques or wavelet methods for solving various problems [10-13]. In contrast, other studies have discussed the applicability of such approaches to approximately treat multiple kinds of integro-differential equations. The application of the Bernstein collocation technique has been extended by Acar [14] to the Stancu collocation technique for the approximate solution of a particular linear integro-differential equation arising from the motion of charged particles. Touchard polynomials and Lagrange Polynomials methods were explored in studies [15-17] for the solution of linear integro-differential equations, including Fredholm, Volterra, and integro-differential equations. Additionally, a novel strategy involving the implementation of the Rationalized Haar wavelet approximate algorithm with the Bernoulli polynomial technique was addressed by Rashid et al. [18]. Several additional investigations, including those using orthogonal basis functions referenced as studies [19-22], have also examined integro-differential equations, including a nonlinear integro-differential equation with a fractional order in complex space. Other works using the operational wavelet matrices of integration can be found in studies [23, 24].

A particular integro-differential equation is called an infinite-boundary integro-differential equation (IBDE) if one or both of its limits are improper. Such equations arise in several important models of tumor-immune system interaction, hematopoiesis, and plankton-nutrient interaction [25], as well as the elastic media problem [26]. Many issues of theoretical physics, electromagnetics, scattering problems, and boundary integral equations lead to IBDE of the second kind in the form,

$u^{\prime}(s)+a(s) u(s)=f(s)+\lambda \int_0^{\infty} k(s, t) u(t) d t$                  (1)

along with the initial condition u(0) = α, where $\lambda \in \mathbb{R}$ is a parameter, both the functions a(s) and f(s) are continuous and the kernel k(s,t) might have a singularity in D = {(s,t): 0 ≤ s, t<∞} while u(s) is the unknown function that needs to be determined. Many researchers have developed an approximate method to solve infinite-boundary integro-differential equations using Galerkin and collocation methods with Laguerre and Hermite polynomials as basis functions [27, 28]. Several studies in the literature have utilized Laguerre polynomial expansions to approximate the IBDEs, particularly in studies [27-34]. However, such approaches have typically focused on evaluating the solution of certain IBDEs with degenerate kernels indirectly rather than solving such problems with any kernel function. Our findings illustrate that the two proposed algorithms yield similar results with fewer orders of generalized Laguerre polynomials, providing many numerical experiments that support the effectiveness depending on the error analysis with different L norms.

This work proposes two novel approximate techniques that combine the generalized Laguerre functions with the penalty optimization algorithm. For the first time, this proposed optimization technique is applied to solve an IBDE given in Eq. (1). The proposed method utilizes a generalized Laguerre series to obtain an approximate solution that converges rapidly and reduces the number of terms that can be computed efficiently. The methodology exhibits computational efficiency, rendering the technique readily implementable on a computer system. Numerous numerical experiments covering a range of applications, including thermal systems, drug accumulation, oscillatory responses, aerodynamic forces, and ecological models, are conducted to assess the efficacy of the proposed methods. These examples show the approach's precision, convergence, and broad applicability. This study presents the following contribution:

• We design a numerical approach that uses the orthogonality of Laguerre polynomials in optimization techniques to solve infinite-boundary integro-differential equations efficiently.
• We present a method of optimization that refines the approximate solutions to increase accuracy and guarantee the satisfaction of boundary constraints. Optimization methods enable the minimization of the error function, resulting in optimal coefficients for the Laguerre polynomial expansion. On the other hand, the penalty method introduces a penalty term into the objective function to ensure that the solution satisfies the given constraints.
• We present MATLAB-based numerical examples, a structured algorithm for implementing the suggested method, and a thorough error analysis to assess accuracy and convergence.
• The proposed approach's accuracy and efficacy in managing intricate integro-differential equations are demonstrated by its validation against known exact solutions. The results illustrate that by selecting the appropriate threshold, the number of nonzero coefficients decreases, and the process speed increases, resulting in an error that is less than a certain amount. Furthermore, the main advantage of our method is its simplicity.

In addition to this introductory section, the remaining part of the paper is outlined as follows: The general form of Laguerre polynomials and their characteristics, such as orthogonality conditions and recurrence relations, are formulated in Section 2. Some new properties are also included in Section 2. The convergence analysis of the proposed approach is presented in Section 3. At the same time, the numerical solution methodology is detailed in Section 4, which includes the formulation of the suggested algorithms and the application of optimization techniques. The numerical results demonstrate the effectiveness and precision of the proposed approaches, which include several experiments as test cases, presented in Section 5. Section 6 concludes the paper with possible avenues for further work.

Under specific conditions, we show mathematically that the approximate solution derived from generalized Laguerre polynomials converges to the exact solution. The analysis demonstrates that the error decreases with increasing order of approximation, utilizing the orthogonality of Laguerre polynomials and the completeness of the Hilbert space. By examining the convergence behaviour, the validity of the suggested approach is verified for resolving integro-differential equations with infinite boundaries. Additionally, this analysis provides guidance on selecting the optimal number of Laguerre polynomials to achieve a balance between accuracy and computational efficiency.

Theorem 3.1: A function u(s) which is continuous on $[0, \infty]$ and satisfied the condition $|u(s)|<M$, M is a constant, can be approximated in terms of generalized Laguerre polynomials as below

$u(s)=\sum_{k=0}^{\infty} c_k L_{g, k}(s)$                (5)

where,

$c_k=\int_0^{\infty} e^{-s} s^n u(s) L_{g, k}(s) d s, k=0,1, \ldots$                  (6)

Then

$u_n(s)=\sum_{k=0}^n c_k L_{g, k}(s)$                    (7)

converges to the function u(s).

Proof. Let $u_n(s)=\sum_{k=0}^n c_k L_{g, k}(s)$, where $c_k=\int_0^{\infty} e^{-s} s^n u(s) L_{g, k}(s) d s$, then

$\int_0^{\infty} e^{-s} s^n u(s) u_n(s) d s=\int_0^{\infty}\left[e^{-s} s^n u(s) \sum_{k=0}^n c_k L_{g, k}(s)\right] d s$               (8)

where, u(s) is defined in Eq. (5). Using Eq. (6) to obtain

$\int_0^{\infty} e^{-s} s^n u(s) u_n(s) d s=\sum_{k=0}^n c_k \int_0^{\infty} e^{-s} s^n u(s) L_{g, k}(s) d s=\sum_{k=0}^n\left|c_k\right|^2$

This means that the function $u_n(s)$ convergence (since it is a Cauchy sequence in the complete Hilbert space $L^2[0, \infty]$).

To prove that the series in Eq. (7) converges to u(s), let

$u_m(s)=\sum_{k=0}^m c_k L_{g, k}(s) d s, for \,\,\, m <n$                 (9)

Hence,

$\begin{aligned} & u_n(s)-u_m(s)=\sum_{k=0}^n c_k L_{g, k}(s) d s-\sum_{k=0}^m c_k L_{g, k}(s) d s & \text { for } n>m\end{aligned}$                 (10)

Eq. (10) leads to $u_n(s)-u_m(s)=\sum_{i=m+1}^n c_k L_{g, k}(s) d s$, for $n>m$.

As a result,

$\left\|u_n(s)-u_m(s)\right\|^2=\sum_{k=m+1}^n\left|c_k\right|^2, n>m$                 (11)

Hence, $\sum_{k=0}^{\infty}\left|c_k\right|^2$ is convergent.

From Eq. (10), one can conclude that $\left\|u_n(s)-u_m(s)\right\|^2 \rightarrow 0$ as $n, m \rightarrow \infty$ and $u_n(s) \rightarrow u(s)$ which is the required result.

In this section, some problems are presented. The obtained results are compared with those of some previous studies. The accuracy of the approximated methods is checked based on the square root of the sum of squared errors norm $L_2$ and the max error norm $L_{\infty}$, respectively:

$\begin{gathered}L_2=\sqrt{\sum_{j=1}^N\left|u_{e x .}\left(s_j\right)-u_n\left(s_j\right)\right|^2} \\ L_{\infty}=\max _j\left[u_{e x .}\left(s_j\right)-u_n\left(s_j\right)\right], j=1,2, \ldots, N\end{gathered}$

where, u_n(s) and u_ex.(s) represent the approximate and exact solutions, respectively. The following examples display the actual and approximate values obtained using both method 1 and method 2 for various test cases with different n. The results are presented in table form. The computations are performed using MATLAB codes.

5.1 Experiment 1: Thermal system with memory response

This model captures historical temperature effects by simulating heat dissipation with memory.

$\begin{gathered}u^{\prime}(s)+u(s)=2+s-2 e^{-s}+\int_0^{\infty} e^{-(t+s)} u(t) d t \\ u(1)=2\end{gathered}$                 (20)

The exact solution is u(s) = 1 + s. This equation can be interpreted as a simplified model of a thermal system with a memory-dependent response in this context:

• The unknown function u(s) represents the temperature of a material or system at a time s.
• The left-hand side u'(s) + u(s) reflects heat loss or energy dissipation typical in conductive or convective systems.
• The nonhomogeneous input 2 + s models an external heat source whose intensity increases over time, such as a gradually heating element.
• The term -2e^-s introduces a localized cooling effect that decays exponentially with time.
• The integral $\int_0^{\infty} e^{-(t+s)} u(t) d t$ Captures a thermal memory kernel, quantifying the influence of past temperature history on the current state.

The computed values of the error norms L_∞ and L₂ are tabulated in Table 1, while the approximate solutions along with absolute errors are plotted in Figure 1 against the exact solution for n = 6.

As seen from Table 1, both the error norms value L_∞ and L₂ are sufficiently small and satisfactorily acceptable, and method 1 produces accurate results that highly match the exact solution. Method 2 gives minimal error (typically < 10^-4), which is acceptable in practical applications and proves its reliability. Furthermore, if the order of generalized Laguerre polynomials increases, the value of the error norms will decrease. That is, better results can be obtained if the value is larger.

Table 1. The norms L_∞ and L₂ for $s \in[1,2,3]$ for Eq. (20)

1a.jpg

(a) Using Algorithm 1

1b.jpg

(b) Using Algorithm 2

Figure 1. The exact and approximate solutions of the thermal system with memory response

5.2 Experiment 2: Drug accumulation model with memory effect

The equation simulates the accumulation of drugs in memory-rich biological systems. Laguerre-based techniques provide a precise, error-free estimate of this experiment over time.

$\begin{gathered}u^{\prime}(s)+u(s)=2 s+s^2-2 e^{-s}+\int_0^{\infty} e^{-(t+s)} u(t) d t \\ \text { with } u(0)=0\end{gathered}$                 (21)

Admits the exact solution u(s) = s², and can be interpreted as a simplified model for drug accumulation in a biological system with a memory effect. This model can be expressed by:

$\int_0^{\infty} e^{-(t+s)} u(t) d t=e^{-s} \int_0^{\infty} e^{-t} u(t) d t$

Acts as a memory operator, capturing past drug concentrations exponentially, weighted cumulative influence. This is particularly relevant in pharmacokinetics, where the systemic impact of a drug depends not only on the current dosage but also on its historical presence in the bloodstream, the differential terms u'(s) + u(s) model the natural decay and metabolic elimination processes of the drug. The nonhomogeneous input 2s + s² – 2e^-s can represent time-dependent admonition or variable absorption dynamics.

Such a formulation is applicable in the release of medication systems, Chemotherapy treatment planning, Long-acting injectable (LAI) administration, Neuropharmacological models involving neurotransmitter uptake, and many other applications. The example in Eq. (21) demonstrates how integro-differential models can be utilized to simulate controlled drug delivery mechanisms with enhanced precision, thereby aiding in the optimization of dosage regimens to maintain therapeutic levels over prolonged periods. The computed values of the error norms value L_∞ and L₂ are tabulated in Table 2.

Table 2. The norms L_∞ and L₂ for $s \in[1,2]$ for Eq. (21)

The approximate solutions, along with absolute errors, are plotted in Figure 2 against the exact solution for n=6.

2a.jpg

(a) Using Algorithm 1

2b.jpg

(b) Using Algorithm 2

Figure 2. The exact and approximate solutions for the drug model with memory effect

5.3 Experiment 3: Oscillatory system with memory response

This oscillatory model incorporates memory effects and sinusoidal input. Increasing the Laguerre polynomial order greatly enhances error reduction and convergence. The following integro-differential equation is used to express this experiment.

$u^{\prime}(s)+u(s)=\cos s+\sin s-0.5 \mathrm{e}^{-\mathrm{s}}+\int_0^{\infty} e^{-(t+s)} u(t) d t$                   (22)

Here, λ = 1 and k(s, t) = e^-(t+s), while the exact solution is u(s) = sin s.

This equation can be interpreted as a simplified model for a dynamical system subjected to a periodic input and exhibiting memory effects. Here's how each component of the equation contributes to the physical interpretation:

• u(s) represents the current state or response of the system at the time s, such as voltage or displacement.
• u'(s) + u(s) is a typical form for first-order linear systems, modeling natural decay or dissipation (e.g., resistance in electrical systems or damping in mechanical systems).
• cos s + sin s corresponds to a sinusoidal input, indicating that a time-varying signal externally drives the system.
• -0.5e^-s acts as a correction or disturbance term that decays rapidly with time, possibly modeling an impulse or initial transient.
• The integral term $\int_0^{\infty} e^{-(t+s)} u(t) d t$ introduces a memory kernel in which the influence of past values of u(t) on the current state is exponentially weighted.

Table 3 shows the computed values of the error norms L_∞ and L₂ using Algorithm 2 for n = 6, 10, 13 and 15. Additionally, the approximate solutions are plotted using the same values and are illustrated in Figure 3. The absolute errors are illustrated in Figure 4. One can show that as n increases, the absolute error approaches zero; this means that the solution converges.

Table 3. The norms L_∞ and L₂ for $s \in[0.1,1.4]$ for Eq. (22)

As n increases, the absolute errors decrease significantly, and the results will rapidly tend to the exact values. The behaviors of the approximate solutions and the corresponding absolute errors for n = 6, 10, 13 and 15. And time step ∆t = 0.1 for $s \in[0.1,1.4]$ are shown in Figures 2 to 3.

As seen from Table 3, the error norms L_∞ and L₂ are sufficiently small and satisfactorily acceptable. Furthermore, it is clear from these tables that as the order of the generalized Laguerre functions n increases, the value of the error norms decreases. In addition, it is confirmed that the sequence of approximate solutions {u_n(s)} converge to the exact solution as n→∞. Mathematically, this means:

$lim _{n \rightarrow \infty}\left\|u(s)-u_n(s)\right\|=0$

The numerical data show:

• Monotonic decrease in maximum absolute error as n increases.
• Uniform convergence over the domain $s \in[0.1,1.4]$, as errors decrease at isolated points and across the entire interval.
• Exponential decay in error, consistent with the spectral convergence property of orthogonal polynomial expansions like Laguerre series, especially for smooth solutions such as sin s. This confirms that the square root of the mean errors will approach zero as n→∞. Figure 3 shows that the approximate solution converges visibly toward the exact curve sin s. Figure 4 illustrates that the absolute error decreases uniformly in the domain for the increasing n, validating global convergence.

3.jpg

Figure 3. The approximate solutions u(s) using the Lageurre polynomial and the exact solution with n = 6, 10, 13, 15 for Eq. (22)

4.jpg

Figure 4. The absolute errors with different n for Eq. (22)

5.4 Experiment 4

In this experiment, a decay system is presented and modeled by the integro-differential Eq. (23). Solving this model using both approaches, we found that it outperforms conventional methods with fewer basis terms and better accuracy.

$u^{\prime}(s)=\frac{1}{5}(-5+2 \sin s-\cos s) e^{-s}+\int_0^{\infty} e^{-(t+s)} \sin (t-s) u(t) d t$                   (23)

Here $\lambda=1$ and $k(s, t)=e^{-(t+s)}$. The exact solution is $u(s)=e^{-s}$. The two methods adopted in section 4 are applied to solve this problem. A comparison of the error norm values obtained by Algorithms 1 and 2 using the order $n=6$ with those which were obtained by the operation matrix of the integration method [27] using the orders $n=10,15$ is illustrated in Table 4. It can be observed from Table 4 that both the error norm values $L_{\infty}$ and $L_2$ obtained by the proposed methods are smaller than those obtained by Nik Long et al. [27]. This means that the present method provides better accuracy with fewer basis functions. The approximate solutions are plotted against the exact ones in Figure 5.

It can be concluded from the numerical results in Table 4, which utilize both the first and second algorithms, as well as the results obtained using the operation matrix of the integration method, that the proposed techniques yield a lower maximum absolute error and are superior to the operation matrix of the integration method with a small order of generalized Laguerre polynomials.

Table 4. The norms L_∞ and L₂ for $s \in[0.1,1.4]$ for Experiment 4

5a.jpg

(a) Algorithm 1

5b.jpg

(b) Algorithm 2

Figure 5. Graph of the absolute errors with n = 6 for Experiment 4

5.5 Experiment 5

For polynomial decay, Laguerre-based solutions correspond to exact values. With much lower absolute errors across the domain, both approaches show excellent performance. Consider the following integro-differential equation:

$u^{\prime}(s)=3 x^2+sin \,\, s-\frac{1}{2}(4+cos \,\,s)+\int_0^{\infty} e^{-(t+s)} sin (t-s) u(t) d t$               (24)

Here λ = 1 and k(s,t) = e^-(t+s) the exact solution is u(s) = s³-2s + 1. The computed values of the error norms L_∞ and L₂ are tabulated in Table 5, while the approximate solutions along with absolute errors are plotted in Figure 6 against the exact solution for n = 6.

6a.jpg

(a) Algorithm 1

6b.jpg

(b) Algorithm 2

Figure 6. Solutions and absolute errors with n = 6 for Experiment 5

Table 5. The norms L_∞ and L₂ for $s \in[0.1,1.4]$ for Eq. (24)

7.jpg

Figure 7. Aerodynamics with wake effects

5.6 Experiment 6: The aerodynamics with wake effects

This model uses wake effects to capture lift and drag. Method two efficiently simulates the impact of historical airflow on present forces by employing penalty constraints. Imagine we have an airplane wing or a flapping wing. We want to calculate how lift and drag (the two main aerodynamic forces) change over time (see Figure 7).

The following system of first-order integro-differential equations is used to define this experiment.

$\begin{aligned} & \acute{u}_1(s)=f_1(s)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}\left(2 s+t^2\right)\left(u_1(t)+u_2(t)\right)\right] d t \\ & \acute{u}_2(s)=f_2(s)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}(t-s)^2\left(u_1(t)-u_2(t)\right)\right] d t\end{aligned}$               (25)

with the initial conditions u₁(0) = 1, u₁(0) = 1 And the exact solutions u₁(s) = s² + 2s + 1, u₂(s) = s²⁺¹.

The unknown functions are: u₁(s) (Lift force at time s and u₂(s) (Drag force at time s). These forces do not depend solely on what is happening at the present moment. They also rely on the history of airflow and vortices, which is called the wake effect, as shown in the following two equations,

Lift equation

$\acute{u}_1(s)=$ local lift effects $+\int_0^{\infty}$ [wake influence on lift] $d t$

Drag equation

$\acute{u}_2(s)=$ local drag effects $+\int_0^{\infty}[$wake influence on drag$] d t$

Using f₁(s), we have

$\begin{aligned} & \acute{u}_1(s)=f_1(s)+\int_0^{\infty}\left[K_1(s)\left(u_1(t)+u_2(t)\right)\right] d t \\ & \acute{u}_2(s)=f_2(s)+\int_0^{\infty}\left[K_2(s)\left(u_1(t)-u_2(t)\right)\right] d t\end{aligned}$

where, $f_1(s)$ and $f_2(s)$ are the instantaneous lift and drag (from the angle, speed, etc.). $K_1(s)$ and $K_2(s)$ are functions that describe how past airflow (wake) influences the current forces, and how the past vortices lose strength over time (due to the $e^{-t}$ terms) and how they interact with the current position s (due to $\left(2 s+t^2\right)$). These mean that when the wing moves, it creates swirling air behind it (vortices), which affects the airflow that hits the wing later, changing the lift and drag. That's why we include the integral terms: they remember the past airflow.

The objective function of the problem is identified for minimization by Algorithm 2. Using Eq. (1), then Eq. (7) and Eq. (18), to obtain

$\begin{aligned} & \sum_{j=o}^n a \acute{L}_j(x)=f_1(x)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}\left(2 x+t^2\right)\left(\sum_{j=o}^n a_j L_j(x)+\sum_{j=o}^n b_j L_j(x)\right)\right] d t \\ & \sum_{j=o}^n b_j \acute{L}_j(x)=f_2(x)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}(t-x)^2\left(\sum_{j=o}^n a_j L_j(x)-\sum_{j=o}^n b_j L_j(x)\right)\right] d t\end{aligned}$

where, $u_1(0)=1, u_2(0)=1$.

The integral of the following error functional generates the first part of the regarded objective function.

$\begin{gathered}\int_0^{\infty}\left\{\left[\sum_{j=o}^n a_j \acute{L}_j(x)-f_1(x)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}\left(2 x+t^2\right)\left(\sum_{j=o}^n a_j L_j(x)+\sum_{j=o}^n b_j L_j(x)\right)\right] d t\right]^2+\right. \\ \left.\sum_{j=o}^n\left[b_j^{\prime} L_j(x)-f_2(x)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}(t-x)^2\left(\sum_{j=o}^n a_j L_j(x)-\sum_{j=o}^n b_j L_j(x)\right)\right] d t\right]^2\right\}\end{gathered}$

If the penalty method is utilized to impose the initial conditions, then

$\begin{gathered}\int_0^{\infty}\left\{\left[\sum_{j=o}^n a_j L_j(x)-f_1(x)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}\left(2 x+t^2\right)\left(\sum_{j=o}^n a_j L_j(x)+\sum_{j=o}^n b_j L_j(x)\right)\right] d t\right]^2+\right. \\ \left.\sum_{j=o}^n\left[b_j^{\prime} L_j(x)-f_2(x)+\int_0^{\infty}\left[t^{\frac{1}{2}} e^{-t}(t-x)^2\left(\sum_{j=o}^n a_j L_j(x)-\sum_{j=o}^n b_j L_j(x)\right)\right] d t\right]^2\right\}+\rho_1\left(\sum_{j=o}^n a_j L_j(0)-1\right)+ \\ \rho_1\left(\sum_{j=o}^n b L_j(0)-1\right)=0\end{gathered}$

where, ρ₁ and ρ₂ are the penalty parameters.

8a.jpg

(a) The solutions

8b.jpg

(b) The absolute error

Figure 8. The results for system (25)

Table 6. The norms L_∞ and L₂ values for $s \in[0.1,1.4]$ for system (25)

The computed values of the error norms L_∞ and L₂ are tabulated in Table 6 with n = 20 and ρ₁ = ρ₂ = 10 while the approximate solutions along with absolute errors are plotted in Figure 8 against the exact solution for n = 6.

5.7 Experiment 7: Ecology with variable limit of integration

An integral limit that varies with time is used to model the dynamics of female populations. This approach yields precise, low-error approximations and effectively manages variable domains. We apply the proposed methods to solve the female population problem in this experiment.

$B^{\prime}(t)=g(t)+\int_0^t k(t, \sigma) B(\sigma) d \sigma$

where, $k(t-\sigma)$, $B(t)$ and $g(t)$ represent respectively the net maternity function of the female class age σ at time t. The number of female births and the contribution of births are already present at the time t.

Let the number of female birth is given by $g(t)=0.25\left(6(1+t)-7 e^{0.5 t}-4 \sin t\right)$, $k(t-\sigma)=t-\sigma, t \in[0,1]$, then the model takes the form as below:

$B^{\prime}(t)-\int_0^t(t-\sigma) B(\sigma) d \sigma=0.25\left(6(1+t)-7 e^{0.5 t}-4 \sin t\right), B(0)=1$                    (26)

The exact solution of the model is $B(t)=0.5\left(e^{0.5 t}-\right. \sin t+\cos t)$.

Table 7 shows the computed values of the error norms L_∞ and L₂ using Algorithm 2 for n = 6, 8, 10. Additionally, the approximate solutions are plotted using the same values and illustrated in Figure 9(a). The absolute errors are illustrated in Figure 9(b). One can show that as n increases, the absolute error approaches zero.

This method creates a new problem, which we solve by reducing the computational effort. Examples demonstrate that our method is better and more flexible than others. Another advantage of this method is its simplicity of application.

Table 7. The norms L_∞ and L₂ for $s \in[0.1,1.4]$ for Eq. (26)

9a.jpg

(a) The solution B(t)

9b.jpg

(b) Absolute errors

Figure 9. The results for Eq. (26) with n = 6, 8 and 10