Haar wavelet collocation method for existence and numerical solutions of fourth-order integro-differential equations with bounded coefficients

1 Introduction

Functional equations, such as partial differential equations (PDEs), integral and integro-differential equations (IDEs), stochastic equations, and others, are typically the outcome of mathematical modelling of real-world issues. IDEs are found in many mathematical formulations of physical processes and are used in chemical kinetics, fluid dynamics, and biological models (for more information, see [1]). IDEs are frequently encountered in problems involving electrostatics, low-frequency electromagnetism, electromagnetism scattering, and the propagation of elastic and acoustic waves. Some applications in scientific disciplines, including microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar range, plasma diagnostics, X-ray radiography, and optical fibre evaluation, when formulated, we obtain in the form of IDEs [2]. Numerous topics are covered in the said area, for instance, the Volterra population growth model, coexisting biological species, the spread of stocked fish in a new lake, heat radiation, and heat transfer. Usually, it is difficult to obtain the exact or analytical solution for every IDEs; therefore, researchers have always tried to establish sophisticated numerical tools to study such problems for their approximate solutions.

Many researchers worked to compute the approximate solution of fourth-order IDEs. Sweilam [3] has used a variational iteration technique (VIT) for the solution of IDEs of order four. Linear and nonlinear fourth-order IDEs have been solved using VIT with boundary conditions. Sweilam et al. [4] have developed a scheme for the solution of fourth-order IDEs based on the pseudo-spectral procedure. Lakestani and Deghan [5] used Chebyshev functions for approximate solutions of fourth-order IDEs. This technique involves expanding the desired solution as Chebyshev cardinal functions. They convert the IDEs to a system of algebraic equations by using an operational matrix. Ghomanjani et al. [6] used the Bezier curve technique to find an approximate solution of fourth-order IDEs. Based on the Adomian decomposition technique, Singh and Wazwaz [7] developed a numerical scheme to address fourth-order boundary value problems (BVPs) of Volterra IDEs. For a recursive method of solution, they applied the Green functions to transform IDEs into integral equations.

The Haar wavelet collocation (HWC) technique is used for different problems in the literature. Some applications of the HWC technique for Lane–Emden equations [8], nonlinear delay IDEs [9], nonlinear delay integral equations [10], software piracy [11], fractional delay differential equations (DEs) [12,13], fractional IDEs [14,15], third-order boundary-value problems of IDEs [16], system of fractional DEs [17], Burger’s equations [18], Schrodinger’s equations [19], inverse problems [20], and interface problems [21]. Majak et al. [22] introduced a higher order Haar wavelet method. Additionally, this technique is used for DEs [23], vibration analysis of beams [24,25], integral equations [26], IDEs [17], nonlinear PDEs [27], and nonlinear evolution equations [28].

Wu et al. [29] used a homotopy-based stochastic finite-element model updating method to cope with the correlation of static measurement data. Cai et al. [30] studies dynamically controlling terahertz wavefronts with cascaded metasurfaces. Yang and Kai [31] studied the nonlinear coupled Schrödinger equation in fibre Bragg gratings. Zhou et al. [32] developed an iterative threshold algorithm for the sparse optimization problems with a log-sum function. Jiang et al. [33] modeled the uncertainty mechanism using an inclusiveness function for each agent to capture the acceptance degree of opposing opinions. The main advantages of this method are its simplicity and less computation costs: it is due to the sparsity of the transform matrices and the small number of significant wavelet coefficients. The Haar method stands out due to its computational efficiency, direct approximation of solutions, ability to handle discontinuities, and flexibility, especially in solving nonlinear IDEs. Its advantages over traditional methods like Laplace, Sumudu, and others make it a powerful tool in numerical analysis, particularly when solving problems require high accuracy with fewer computational resources [34].

Existence theory is an important consequence of applied analysis. The said analysis is used to investigate whether a particular problem that is under investigation has a solution or not. For this purpose, an important theory was founded by Banach in 1922. After him, many theories and results were established to study the existence theory of solutions for integral, functional, algebraic, and DEs. Previous studies [35–37] have valuable contributions to the theory of existence of solutions. This theory tells us whether the problem has a solution or not. If yes, how many solutions are there for a particular problem? Therefore, fixed point theory and nonlinear analysis tools like the degree theories of Mawhin and Schauder have gained much attention in the last few decades. We refer to some work like [38–40]. Here, we remark that the existence theory of IDEs has also been considered recently by many researchers in their work. For instance, Shatanawii et al. [41] have studied the existence of solutions to some integral problems. In the same way, Shatanawii et al. [42] and Isik et al. [43] established the existence theory for some IDE problems.

In this study, fourth-order linear Volterra–Fredholm IDEs are solved using the HWC technique. Consider the subsequent fourth-order linear IDE:

subject to

where a 1 , a 2 , a 3 , a 4 , f are known functions, ξ is an unknown function, k 1 , k 2 : [ 0 , 1 ] × [ 0 , 1 ] → R are defined, t and s functions called as the integral’s kernel, μ 1 , μ 2 , λ 1 , λ 2 , λ 3 , and λ 4 are any constant real numbers. In addition, the following fourth-order nonlinear Volterra–Fredholm IDEs are solved using the HWC technique as

subject to

here F : [ 0 , 1 ] × R → R is a nonlinear continuous function. Inspired by the importance of fixed point theory, here we establish a useful result based on Banach fixed point to verify the uniqueness and existence of solutions to the proposed problem.

The structure of this article is as follows. Section 2 addresses Haar functions, and Section 3 presents existence and uniqueness results. The numerical scheme for solving nonlinear fourth-order IDEs is provided in Section 4. In Section 5, convergence is provided. Examples from the literature are provided in Section 6. Section 7 presents the results and comments. Section 8 provides the conclusion.

2 Haar wavelet

The scaling function is [9]

The Mother wavelet is given by

The dilation and translation operations provide h 2 ( t ) , from which all functions in this family are formed. These functions are defined on subintervals of [0, 1). The scaling function can be stated as follows, but all other functions are specified for t ∈ [ a , b )

where α = k m , β = k + 1 ∕ 2 m , γ = k + 1 m , for i = 2 , 3 , … , 2 M = N , m = 2 j , where k = 0 , 1 , 2 , … , m − 1 and j = 0 , 1 , 2 , … , J , J = 2 M . Translation parameter is K , while the wavelet’s level is indicated by the number j , integer J represents the highest resolution level. 1 + k + m = i is relation between i , m , and k . Expression for every square integrable f ( t ) in [0, 1) is

For purposes of approximation, this is truncated at N terms as

we use the symbol

and the definition of h i is used to determine the value of the aforementioned integral, which is provided by

Through integral simplification, we can obtain the values of P i , 2 ( t ) , P i , 3 ( t ) , and P i , 4 ( t ) .

The value of P i , n is

If i = 1 , then

The interval I is divided into subintervals as

The points t j ’s are known as collocation points (CPs). Gauss points (GPs) are defined as

3 Existence and uniqueness results

This section is related to existing results. Let denotes X = C 4 [0, 1] Banach space, with norm as

We make some assumptions that will be used in the main result:

1. Since a 1 , a 2 , a 3 , a 4 , and f are continuous, so there exist a * , b * , c * , d * , f * > 0 such that

   ∣ a 1 ( t ) ∣ ≤ a * , ∣ a 2 ( t ) ∣ ≤ b * , ∣ a 3 ( t ) ∣ ≤ c * , ∣ a 4 ( t ) ∣ ≤ d * , ∣ f ( t ) ∣ ≤ f * .

2. k 1 , k 2 : [ 0 , 1 ] × [ 0 , 1 ] → R are continuous and bounded, so ∃ k 1 * , k 2 * > 0 such that

   ∣ k 1 ( t , s ) ∣ ≤ k 1 * , ∣ k 2 ( t , s ) ∣ ≤ k 2 * .

3. Since F is a continuous function and satisfies the Liptchitz condition, such that for any ξ , ν ∈ X , one has

   ∣ F ( t , ξ ( t ) ) − F ( t , ν ( t ) ) ∣ ≤ L F ∣ ξ − ν ∣ , L F > 0 .

Further, we write (13) as

The solution of nonlinear problem Eq. (2) can be written in equivalent integral form as

Note: Since we see that max t ∈ [ 0 , 1 ] 1 24 , 1 6 , 1 3 , 1 2 = 1 2 , we state the given theorem.

4 Numerical method

Here, HWC technique is developed for the solution of fourth-order Volterra–Fredholm IDEs. The integration method yields expression for lower derivatives, whereas the Haar function approximates the fourth-order derivative. Additionally, formula (30) is used to obtain the integral involved in Eq. (1)

We can obtain the algebraic equations system by placing the CPs and GPs in Eq. (1) and using the HWC technique to it. The unknown Haar coefficients are found using the Gauss elimination method. Ultimately, solution at the CPs is obtained by use of these coefficients. Consider ξ ( 4 ) ( t ) ∈ L 2 [ 0 , 1 ) , so ξ ( 4 ) ( t ) is written as

Integrating from 0 to t , we obtain

Using the initial condition, we have

Again integrating from 0 to t , we have

After establishing an initial condition, we have

Again integrating from 0 to t , we have

Again using the initial condition, we have

Integrating from 0 to t , we obtain

Using the initial conditions and rearranging, we have

Eq. (34) is known as numerical solution.