Abstract
We present a new numerical technique to discover a new solution of Singular Nonlinear Volterra Integral Equations (SNVIE). The considered technique utilizes the Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet method (HOBW) to solve the weakly SNVIE including Abel’s equations. We acquire the HOBW implementation matrix of the integration to derive the procedure of solving these kind integral equations. The explained technique is delineated with two numerical cases to demonstrate the benefit of the technique used by us. At last, the exchange uncovers the way that the strategy utilized here is basic in usage.
1. Introduction
In the current literature, there are many different applications of SNVIE in various areas, such as mathematical physics, electrochemistry, scattering theory, heat conduction, semiconductors, population dynamics, and fluid flow [1, 2]. Numerical strategies for the SNVIE are spline collocation methods [3], Newton–Cotes methods [4], extrapolation algorithm [5], and Hermite-collocation method [6]. The most popular methods for talking about the such equations are introduced, such as homotopy asymptotic method [7], Nyström interpolant method [8], Mesh method [9], Tau method [10], Laplace transform [11], orthonormal Bernstein, and block-pulse functions [12–17].
Wavelet theory is a moderately new and considered as a rising territory in the field of applied science and engineering. Wavelets allow the accurate representation of a lot of functions. The wavelet technique is a new numerical technique utilized for dissolving the fractional equations. SNVIE has numerous applications in different zones, for example, semiconductors’ mathematical chemistry, chemical reactions, physics, scattering theory, electrochemistry, seismology, metallurgy, fluid flow, and population dynamics [2, 18–20].
In 1823, Niels Henrik Abel derived the equationwhere is an unknown function and is a given function. This equation is an example of a nonhomogeneous Volterra equation of first kind with weak singularity. Abel obtained this equation while studying the motion of a particle on a smooth curve lying on a vertical plane. The physical depiction of this condition is given in [21] as pursues. Abel thought about the issue in traditional mechanics, which is that of deciding the time a molecule brings to slide openly down a smooth settled bend in a vertical xy-plane (in Figure 1), from any settled point on the bend to its absolute bottom (the starting point 0). If means the mass of the molecule and signifies the condition of the smooth bend where is a differentiable function of , at that point we acquire the vitality protection condition aswhere is the speed of the molecule at the position at time , if the molecule tumbles from rest at time from the point , and represents acceleration due to gravity. The connection (2) can be expressed asby utilizing the arc-length , estimated from the starting point to the point, where a less sign has been utilized in the square root since diminishes with time amid the fall of the molecule. Using the formulawe can composeBy integrating both sides of (5), we obtain where is the total time of fall of the particle, from the point to the origin Therefore, we havewhere . In this way, we can find that the time of descent of the particle, T, can be resolved totally by utilizing the recipe (7), if the state of the curve , and consequently the function is known. On the off chance that we consider, on the other hand, the issue of assurance of the state of the bend, when the time of fall is known, which is the historic Abel’s problem, then the relation (7) is an integral equation for the unknown function , which is known as Abel’s integral equation.
The most general form of Abel’s integral equation is given bywhere is a monotonically expanding function. We have picked it as . Also, a general form of SVIE of second kind is given aswhere is in on the interim and is a steady parameter.
We utilize the HOBW method for determining the approximation solution of SNVIE of the shape given bywhere are continuous functions, while and is the unknown function to be determined.
This paper is organized as follows. Initially the basic formulation of the HOBW method and some properties of HOBW are defined in Section 2. In Section 3, we determine the HOBW implementation matrix of integration. While in Section 4, we summarize the process of dissolving weakly singular-Volterra integral equations based on the HOBW implementation matrix method. In Section 5, we consider two examples which demonstrate the validity of this method. Finally, the concluding remarks are demonstrated.
2. The HOBW Method and Operational Matrix of the Integration
2.1. Wavelets and the HOBW Method
Wavelets constitute a group of functions constructed from dilation and translation of a single function called the mother wavelet. In which parameter of dilation and parameter of translation vary continuously.By letting and be discrete values such as , ,
where and are positive integers, we attain the family of discrete wavelets:Then we see that forms a wavelet basis for . In particular, when , , then forms an orthonormal basis. Here, involves four arguments, is to be any positive integer, is the degree of the Bernstein polynomials, and is the normalized time. are defined on [0, 1) as [12, 13]where , and is a positive integer. Thus, we attain our new basis as and any function is truncated with them.
The detect orthonormal basis is given by where is called the inner product in . The has compact support , .
2.2. Function Approximation by Using the HOBW Functions
Any function , which is integrable in , is truncated by using the HOBW method as follows:where the HOBW coefficients can be calculated as given below:We approximate by a truncated series as follows:where and are vectors given byandWe define the HOBW matrix as follows:The series in (17) contains an infinite number of terms for a smooth function . Therefore, we haveso thatwhereThen, by using (14), is defined as follows:We can also approximate the function as follows:where is an matrix that we attain as follows:
2.3. Multiplication of the Hybrid Functions
We can evaluate for VIE of the second kind via the HOBW functions as detailed below.
Let the product of and be given bywhereWith the recursive formulas, we calculate for any and .
The matrix in (23) satisfies the following relation:where is defined in (33) and is the matrix coefficient. We consider the case when and . Thus, we haveThe coefficient matrix in (33) is determined bywhere are matrices given by
3. HOBW Operational Matrix
Firstly, we review some basic definitions of fractional calculus [22–24], which are required for establishing our results.
Definition 1. The Riemann–Liouville fractional integral operator of order , of a function , , is defined as follows:The block-pulse functions (BPFs), an -set of BPFs on [0, 1), are defined bywhere The BPFs have the orthogonal properties as follows:andEvery function which is integrable in can be truncated with the aid of BPFs series aswhere .
Using the disjointness of BPFs and the matrix of can be gotten byEquation (41) implies that the HOBW method can be truncated into an -set BPFs as follows:The block-pulse implementation matrix of the fractional integration has been given in [14] as follows:whereAt , is BPF’s implementation matrix of integration.
Letwhere the matrix is called the HOBW implementation matrix of fractional integration [2, 17]. Using (43) and (44), we have From (38) and (39) we can getThen the matrix is given byFor example, when , M = 2, and = 3, the operational matrix of the fractional integration is expressed as follows:
4. Solution of Nonlinear Volterra Integral Equations via the HOBW Method
Consider the following integral equation:where is matrix with The functions can be truncated into the HOBB functions asTherefore, upon substituting into (52), we getwhere With the aid of the previous equations, (52) becomes where .
To compute the unknown HOBW coefficients, we use the collocation points as follows:From (60), we have a system of nonlinear equations with unknowns. Newton iteration method is used for completing the solution of the resulting nonlinear system, to get the unknown vectors . So, the approximated results can be calculated as
5. Numerical Examples
We use the demonstrated technique in this article for finding the numerical results of four weakly singular-Volterra integral equations.
Example 1. Consider the generalized Abel’s integral equation [21].The exact solution is .
The outcomes demonstrate the high exactness and the effectiveness of the technique. This outcome can be effortlessly confirmed that the strategy yields the desired accuracy only in a few values of and . The results of this example at different values of k and M are presented in Table 1.
Example 2. Consider the following WSVIE:The exact solution is .
Table 2 likewise checks all favorable circumstances of the strategy examined in the past examinations. It ought to be noticed that the HOBW additionally effortlessly composes PC code. This is another vital trademark for the numerical calculation. These actualities delineate the HOBW strategy as a quick, dependable, legitimate, and useful asset for understanding WSVIEs.
Example 3. Consider the singular kernel Volterra integral equation [25]:The analytic solution of (49) can be detected in [18] as
The comparison among the solution and the second Chebyshev wavelet (SCW) solution is shown in Table 3 for and , which confirms that the method gives almost the closer loose as the analytic solution. Figure 2 shows the comparison among the HOBW solution and the analytic one for . Better approximation is expected by the values of and as in Table 2.
Example 4. Consider the nonlinear Volterra integral equation with singular kernel [25]:with the exact solution .
The comparison among the HOBW solution and the analytic solution for is shown in Table 4 and Figure 3 for and and confirms that the HOBW method gives almost the same solution as the analytic method. Better approximation is expected by choosing higher values of and .
6. Conclusion
In this investigation, the combination of orthonormal Bernstein, block-pulse functions, and wavelets is applied for resolving SNVIE. The main purpose of our method is to combine the orthonormal Bernstein and block-pulse functions wavelet method with the definition of the Riemann–Liouville fractional integral with the singular integral. The method depends on reducing the considered system to a set of nonlinear algebraic equations. The generated system just needs sampling of functions and no integration. Wavelets as orthogonal systems have different resolution capability for truncating functions by the increasing of dilation parameter that can give a good truncation for integral equations without using a polynomial solution. The considered method has its efficiency and simplicity. The matrices D and P are sparse; hence the CPU time and the computer memory will be reduced and at the same time the solution remains accurate. We also noted that when the degree of HOBW is increased, the errors will be decreased to smaller values. When the values of and are higher, we get more accurate solutions for the given problems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project Number No. 20-1440. They would also like thank the editing board and reviewers for their valuable response and fast reply that enhance the obtained results.