The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation

ABSTRACT

There are two methods for solving integral equations: the analytical and the numerical methods. In previous research, Abel's integral equations of the first kind ( = 1 2 ) is solved analytically with Laplace transform (Aggarwal & Sharma, 2019). Meanwhile, generalizations of Abel's integral equations of the first kind (0 < < 1) are solved analytically using fractional calculus (Jahanshahi et al., 2015) and fractional-order Mikusinski operator (M. Li & Zhao, 2013). Solving the Abel's integral equations with numerical methods has been carried out using Touchard and Laguerre polynomials (Abdullah et al., 2021) and the Laguerre wavelet (Mundewadi, 2019). In addition, the generalization of Abel's integral equations has been solved by numerical methods, named the hp-version collocation method (Dehbozorgi & Nedaiasl, 2020). This study will determine the solutions of generalization of the first and second kinds of Abel's integral equations. The methods that will be used in this study include analytical methods, which include Laplace transforms, fractional calculus, and equation manipulation. In previous study, the Laplace transform method was used by Aggarwal & Sharma (2019) to solve the first kind of Abel's integral equation ( =   1   2 ). In this paper, the ideas from Aggarwal & Sharma (2019) will be used to determine the generalization solutions of the first and second kinds of Abel's integral equations (0 < < 1). Determination of generalization solutions with fractional calculus using ideas from (Jahanshahi et al., 2015). This study also uses numerical methods when solutions cannot be determined analytically. The numerical method chosen is successive approximations. After obtaining the generalization solution, it is expected to be able to solve various problems (stereology, seismology, geometry, etc.), which are represented in the generalized form of the first or second kind of Abel's integral equations.

B. METHODS
This research studies the generalization of the first and second kinds of Abel's integral equations, and then the solution will be determined. The first step is to learn some of the analytical methods used to determine the solution of the integral equation. Next, select the methods that can be applied to Abel's integral equation, including the Laplace transform, fractional calculus, and equation manipulation. When the solution of the equation cannot be determined by analytical methods, the solution is determined by numerical methods. In this study, the numerical method chosen was successive approximations.

Fractional Calculus
Fractional calculus is a branch of classical mathematics which deals with the generalization of operations of differentiation and integration to fractional order (Delkhosh, 2013). Thederivative of a function ( ) is ( ) . In general, the value of is a positive integer, can be a rational, irrational, or complex number.
Definition of fractional integral by Riemann-Liouville: and the Riemann-Liouville fractional derivative is defined as: (8)

C. RESULT AND DISCUSSION 1. Generalization of Abel's Integral Equations of the First Kind
Let be any known function, and be the function to be determined. The Abel's integral equation of the first kind is Equation (9) can be generalized as follows: where ( , ) = 1 ( − ) is called an Abel's kernel. In this case, equation (9) is a special case of generalization of the first kind of Abel's integral equation with a value of = 1 2 .
A further generalization of first kind of Abel's integral equations considers the following Abel's kernel: equation (10) becomes where ( ) is a a monotonic function of increasing and decreasing in the interval 0 < < and ′ ( ) ≠ 0 for every in the interval.

Theorem 2
Let be any known function, with

Proof:
There are three methods to prove Theorem 2.
a. Laplace Transform The proof of the Laplace transformation uses ideas from Aggarwal & Sharma (2019). The first step is to use the definition of convolution for equation (10) so that Then a Laplace transform is performed on both sides of equation (12) ℒ{ where Γ is the Gamma function. To obtain the inverse Laplace transform, equation (15) can be expressed by By applying the inverse of the Laplace transform (ℒ −1 ) to equation (19), we get

b. Equation Manipulation
Both sides of equation (10)  (23) Furthermore, both sides are derived from the variable so that (24) The variable , is a dummy variable, then

c. Fractional Calculus
The proof of the fractional calculus uses ideas from Jahanshahi et al. (2015). Based on the definition of fractional integral by Riemann-Liouville, equation (10)  From equation (28), it is obtained that (30) Then both sides are derived to , then The variable , is a dummy variable, so it applies

Generalization of Abel's Integral Equations of the Second Kind
Let be any known function, and be the function to be determined. The Abel's integral equation of the second kind is where ( ) is a monotonic function of increasing and decreasing in the interval 0 < < and ′ ( ) ≠ 0 for each in the interval.

Proof:
The Laplace transform method is used to prove Theorem 4. By carrying out the Laplace transformation on both sides of equation (33), it is obtained that where Γ is a gamma function, ( ) = ℒ{ ( )}, and ( ) = ℒ{ ( )}. The inverse of the Laplace transforms (ℒ −1 ) for both sides is The solution ( ) in Theorem 4 is still in the inverse Laplace form, so it is difficult for some functions . Therefore, the equation (33) solution can also be approached using numerical methods, namely successive approximations. This method begins by determining the initial guess. Then the initial guess is an approximation of the next function (Kanwal, 2013). The recurrence relation of equation (33) for this method is When the initial guess 0 ( ) = 0, some successive approximations with ≥ 1 are

Illustration of Abel's Integral Equation
The d. According to Thórisdóttir & Kiderlen (2013), the first kind of Abel's integral equation is applied in stereology, namely in the random ball model, with the equation is ( 1 ): spherical particle size distribution ( 2 ) : circular section size distribution of the random plane section particle : upper limit on the maximum size of spherical particles : average radius of the sphere.

D. CONCLUSION AND SUGGESTIONS
Suppose is any known function, and is a known increasing and decreasing monotone function, and is the function to be determined, then: The generalization of the first kind of Abel's integral equation solution using the Laplace transform method, fractional calculus, and equation manipulation is The three analytical methods cannot determine further generalization of the second kind of Abel's integral equations solution, so determined by numerical methods (successive approximations). The solution is ( ) = lim →∞ +1 ( ). In this study, the further generalization solution of the second kind of Abel's integral equation is determined using a numerical method, namely successive approximations. Future research is expected to determine the advanced generalization solution of the second kind of Abel's integral equation using analytical methods.