Using Hermite Polynomials to Solve Volterra-Fredholm Integral Equation of the Second Kind

The integral equations form the basis of several mathematical models in many scientific domains, such as engineering, mathematics, and chemistry. Numerous areas in engineering and applied mathematics are connected to integral equations as defined in the study conducted by Wazwaz [1] and Semenova et al. [2].

The Volterra-Fredholm Integral Equations have played a major role in developing many crucial in applied mathematics, engineering, and physics [3], and have found extensive application in various scientific fields. The Volterra and Fredholm integral equations are combined to create these integral equations.

We define the Linear Volterra-Fredholm Integral Equations of the second kind (LVFIES for short) as follows [3-5]:

$\begin{gathered}\lambda(\tau)-\int_{-1}^\tau h_1(\tau, v) \lambda(v) d v-\int_{-1}^1 h_2(\tau, v) \lambda(v) d v=f(\tau) \\ -1 \leq \tau, v \leq 1\end{gathered}$     (1)

In the case where f(τ), h₁(τ, ν) and h₂(τ, ν) are functions. The function λ(τ) is the unknown function that must be determined. There are many techniques for properly effectively resolving such problems [6, 7], such as Adomian's method [8, 9], Chebyshev polynomials [10-12], They serviced the Volterra Friedolm integral equation using a Chebychev polynomial with summation points. Then they proposed a new summation method to solve the Volterra Fredholm equation, Euler series to solve the Fredholm integral equation [13], Laplace transform method [14, 15].

In this context, Salman Mustfaf [16] presented an innovative method for solving linear fractional Volterra-Fredholm Integro-Differential Equations (LFVFIDEs) with fractional derivatives in the Caputo sense. To validate the effectiveness of this new approach, it is compared with existing techniques, demonstrating its superior performance in addressing these complex problems. Varol et al. [17] analyzed the application of Laguerre polynomials in constructing a numerical approximation method for solving fractional linear integral-differential equations (IDEs) of the Fredholm-Volterra type.

Negarchi and Nouri [18], and Nemati [19] discussed solutions to the integral equation using the Legendre polynomial.

In our work, we use Hermite polynomials to solve the Volterra-Fredholm equation integral. In the context of numerical methods, convergence analysis is crucial to ensure that the method reliably approximates the exact solution as the number of terms increases.

The paper’s structure is: introductory section contains an introduction to the LVFIES, in the second one, we demonstrate the existence and the uniqueness of the solution of the LVFIES, and then we introduce definitions about the Hermite polynomials, which is the basis of our work in the third section. In the fourth one, we put the description of the collocation Hermite method to solve LVFIES we search for an approximate solution in the form of series. In the fifth section, numerical examples are presented to demonstrate the accuracy and efficiency of the current method. Finally, the conclusions of this work are summarized.

Hermite polynomials are a series of orthogonal polynomials that appear in probability theory, physics, and numerical analysis.

Hermite polynomials are defined by its general form [20, 21]:

$H t_n(\tau)=(-1)^n e^{\frac{\tau^2}{2}} \frac{d^n}{d \tau^n} e^{-\frac{\tau^2}{2}} \quad, n=0,1,2 \ldots ,-\infty<\tau<+\infty$

The first seven Hermite polynomials are:

$\begin{aligned} & H t_0(\tau)=1, \\ & H t_1(\tau)=2 \tau, \\ & H t_2(\tau)=4 \tau^2-2, \\ & H t_3(\tau)=8 \tau^3-12 \tau, \\ & H t_4(\tau)=16 \tau^4-48 \tau^2+12, \\ & H t_5(\tau)=32 \tau^5-160 \tau^3+120 \tau, \\ & H t_6(\tau)=64 \tau^6-480 \tau^4+720 \tau^2-120, \end{aligned}$

There is orthogonality in the polynomials Ht_n(τ) with respect to the weight function $S(\tau)=e^{-\frac{\tau^2}{2}}$ with the following condition:

$\int_{-\infty}^{+\infty} H t_n(\tau) H t_m(\tau) S(\tau) d \tau=n!\sqrt{2 \pi} \delta_{n, m}$     (2)

δ_n,m denoted the Kronecker delta.

Let:

$\omega_n(\tau)=\frac{1}{\sqrt{n!\sqrt{2 \pi}}} \times H t_n$      (3)

So:

$\left\langle\omega_n, \omega_m\right\rangle=$$\frac{1}{n!\sqrt{2 \pi}} \int_{-\infty}^{+\infty} H t_n(\tau) H t_m(\tau) S(\tau) d \tau=\left\{\begin{array}{l}1 \text { si } n=m \\ 0 \text { si } n \neq m\end{array}\right.$      (4)

In this part, we give some illustrative examples to clarify our work

Example 1

Consider the LVFIES

$\lambda(\tau)-\int_0^\tau\left(2 \tau^2 v+1\right) \lambda(v) d v-\int_0^1 \tau(v+1) \lambda(v) d v=f(\tau)$

If we take:

$f(\tau)=-\frac{2}{5} \tau^7-\frac{5}{4} \tau^4+\tau^3-\frac{59}{20} \tau+1$

We get:

$\lambda(\tau)=\tau^3+1$

Applying the Hermite collocation method to approximate the solution λ(τ).

Table 1. Results for Example 1

1.png

Figure 1. The exact and the approximate solutions for Example 1 with n=3

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Figure 2. The absolute errors for Example 1

In Table 1, we present the error for Example 1 for N=3 then comparing the results with one of Pull Lucas method [25].

Figures 1 and 2 present respectively comparisons of the exact and approximate solution for n=3, and the absolute errors for Example 1.

Example 2

Consider the LVFIES

$\lambda(\tau)-\int_{-1}^\tau e^{-(\tau+v)} \lambda(v) d v-\int_0^1 \lambda(v) d v=f(\tau)$

If we take:

$f(\tau)=e^{-\tau}+e^{-\frac{3 \tau}{2}}+e^{-1}-e^1-e^{\frac{2-\tau}{2}}$

We get:

$\lambda(\tau)=e^{-\tau}$

Table 2. Comparison of the errors for Example 1

In Table 2, a comparison of the exact and the approximate solution for N=8, N=10 is done.

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Figure 3. Comparison of the exact and approximate solutions for Example 2

In Figure 3, we depicted the exact solution and the approximate solution for n=8 of the problem in Example 2 obtained by Hermit collocation method.

Example 3

Consider the LVFIES

$\lambda(\tau)-\int_{-1}^\tau e^{-(\tau+v)} \lambda(v) d v-\int_0^1 2 \cosh \lambda(v) d v=f(\tau)$

If we take:

$\begin{aligned} f(\tau) & =\frac{70368744177664}{325586356356385549} \cosh (\tau)-\frac{70368744177664}{325586356356385549}(\cosh (\tau)(\sin (2)+2)) \\ & +\frac{70368744177664}{325586356356385549} \tau^*\left(e^{(-\tau) / 2}-e^{-1}+e^{\tau / 2}+r\left(e^{-\tau / 2}-e^{\tau / 2}\right)\right)\end{aligned}$

We get:

$\lambda(\tau)=\frac{\cosh \tau}{\sinh 2 \tau+1}$

Table 3. Comparison of errors for Example 3

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Figure 4. The absolute errors for Example 3

The comparison of the error of Example 3 is presented in Table 3, according to different values of N such that 6, 8 and 10. It can be noted that the absolute error decreasing as the value of the integer N increasing.

Results presented in the previous table are shown in the Figure 4, we remark that error decreases when N increases.

Example 4

Consider the LVFIES:

$\lambda(\tau)-\int_{-1}^\tau \cos (\tau-v) \lambda(v) d v-\int_{-1}^1 \sin (\tau-v) \lambda(v) d v=f(\tau)$

If we take:

$f(\tau)=e^{\tau / 2}+e^{-1}(\cos (\tau+1)-\sin (\tau+1))$$+e^{-1}(\cos (\tau+1)+\sin (\tau+1))$$-e^1(\cos (\tau-1)+\sin (\tau-1)) / 2$

We get:

$\lambda(\tau)=e^\tau$

Table 4 presents the comparison of the error of Example 4 for N=8 and N=10.

In Figure 5, we depicted the exact solution and the approximate solution for n=6 of the problem in Example 4 obtained by Hermit collocation method.

Table 4. Comparison of the errors for Example 4

5.png

Figure 5. The absolute errors for Example 4

Example 5

Consider the LVFIES

$\lambda(\tau)-\int_{-1}^\tau(\tau-v) \lambda(v) d v-\int_{-1}^1 \tau \lambda(v) d v=f(\tau)$

If we take:

$f(\tau)=2 e^\tau-5 e^{-1}+e^{-1}-4 \tau e^{-1}\left(e^2-1\right)$

We get:

$\lambda(\tau)=\tau e^\tau$

Table 5. Results for Example 5

The comparison of the exact and the approximate solutions for N=10. The results for Example 5 are presented in Table 5.

Example 6

Consider the LVFIES

$\lambda(\tau)-\int_0^\tau v \lambda(v) d v-\int_{-1}^1 v \lambda(v) d v=f(\tau)$

If we take:

$f(\tau)=3 \tau+4 \tau^2-\tau^3+\tau^4-2$

We get:

$f(\tau)=3 \tau+4 \tau^2$

The comparison of the exact and approximate solutions for N=3 for Example 5 is presented in Figure 6.

Figures 7 and 8 present respectively the absolute errors for Example 6 and the absolute errors for Example 7. The results for Example 5 are presented in Table 6.

6.png

Figure 6. The exact and approximate solutions for Example 5

7.png

Figure 7. The absolute errors for Example 6

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Figure 8. The absolute errors for Example 7

Table 6. Results for Example 6

Example 7

Consider the LVFIES

$\lambda(\tau)-\int_0^\tau(\tau-v) \lambda(v) d v-\int_0^1 \tau \lambda(v) d v=f(\tau)$

If we take:

$f(\tau)=-2-2 \tau+e^\tau$

We get:

$\lambda(\tau)=\tau e^\tau$

The comparison of the exact and the approximate solutions for N=4 and N=8. Tables 7 and 8 present respectively the results for Example 7 and a comparison of errors for Example 7 with Adomian. Figure 8 presents the absolute errors for all errors of the same example.

Table 7. Results for Example 7

Table 8. Comparison of errors for Example 7 with Adomian

Convergence analysis

Hermite polynomials, owing to their orthogonality and recurrence relations, exhibit strong convergence properties when applied to functions that can be well-represented.

Error bounds

Error bounds provide a quantitative measure of how close the numerical solution is to the exact solution. For Hermite polynomials, the error bound is influenced by the smoothness of the target function and its behavior at the boundaries. This error bound illustrates that as N increases, the error decreases factorially, indicating high accuracy for sufficiently large N.

Stability

Stability is a critical aspect in numerical methods to ensure that errors do not amplify through computations. Hermite polynomials benefit from their orthogonality, which inherently contributes to numerical stability. This property ensures that the polynomial terms do not interact destructively, maintaining the integrity of the approximation over a wide range of values.

Moreover, the stability of Hermite polynomial-based methods is reinforced by their recurrence relations, which allow for the efficient and stable computation of higher-order terms. These relations mitigate the risk of numerical instability that might arise from direct computation methods, making Hermite polynomials a robust choice for high-precision applications.