Solution of Nonlinear Volterra Integral Equations with Weakly Singular Kernel by Using the HOBWMethod

1Department of Mathematics, Benha Faculty of Engineering, Benha University, Benha, Egypt 2Department of Mathematics, College of Sciences and Human Studies at Howtat Sudair, Majmaah University, Al–Majmaah 11952, Saudi Arabia 3Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA 4Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia 5College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China 6International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

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][19][20].
In 1823, Niels Henrik Abel derived the equation where () 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 as where V is the speed of the molecule at the position (, ) at time , if the molecule tumbles from rest at time  = 0 from the point (, ), and  represents acceleration due to gravity.
The connection (2) can be expressed as by 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 formula we can compose By integrating both sides of (5), we obtain where  is the total time of fall of the particle, from the point (, ) to the origin (0, 0).Therefore, we have where (0) = 0.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 by where ℎ() is a monotonically expanding function.We have picked it as ℎ().Also, a general form of SVIE of second kind is given as where () is in  2 () on the interim  ≤ , 0 ≤  and  is a steady parameter.We utilize the HOBW method for determining the approximation solution of SNVIE of the shape given by  () =  () +  ∫    (, )  ( ()) ( − ) 1− , 0 ≤  ≤ 1. (10) where (), (, ) are continuous functions, while 0 <  < 1 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.
The detect orthonormal basis is given by where

HOBW Operational Matrix
Firstly, we review some basic definitions of fractional calculus [22][23][24], which are required for establishing our results.
Definition 1.The Riemann-Liouville fractional integral operator  of order , of a function  ∈  V , V ≥ −1, is defined as follows: The block-pulse functions (BPFs), an -set of BPFs on [0, 1), are defined by where  = 0, 1, 2, . . .,  − 1.The BPFs have the orthogonal properties as follows: and Every function () which is integrable in [0, 1) can be truncated with the aid of BPFs series as where Using the disjointness of BPFs and the matrix of   () can be gotten by Equation ( 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: where At  = 1,   is BPF's implementation matrix of integration.Let where the matrix P  2 −1 ×2 −1  is called the HOBW implementation matrix of fractional integration [2,17].Using ( 43) and (44), we have (48) From ( 38) and (39) we can get Then the matrix is P  2 −1 ×2 −1  given by For example, when  = 0.5, M = 2, and  = 3, the operational matrix of the fractional integration P  2 −1 ×2 −1  is expressed as follows: [ 0.19343
The functions   () can be truncated into the HOBB functions as Therefore, upon substituting into (52), we get where  7.1 × 10 −5 0.9 0.9791483643 0.9791483624 1.9 × 10 −9 6.9 × 10 −5 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 2 −1  nonlinear equations with 2 −1  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

Numerical Examples
We use the demonstrated technique in this article for finding the numerical results of four weakly singular-Volterra integral equations.
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.
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.The analytic solution of (49) can be detected in [18] as () =  3 .
The comparison among the  solution and the second Chebyshev wavelet (SCW) solution is shown in Table 3 for  = 4 and  = 2, 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  ∈ [0, 1).Better approximation is expected by the values of  and  as in Table 2.The comparison among the HOBW solution and the analytic solution for  ∈ [0, 1) is shown in Table 4 and Figure 3 for  = 4 and  = 2 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 .

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.

Table 1 :
Maximum absolute errors at different values of  and  for Example 1 via HOBW.

Table 2 :
The comparison among HOBW, exact, and Chebyshev solutions for Example 2.

Table 3 :
The comparison among HOBW, exact, and SCW solutions for Example 3.

Table 4 :
The comparison among HOBW, analytic, and SCW solutions for Example 4.