Open AccessArticle

Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients

Chenkuan Li

and

Hunter Plowman

Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada

Author to whom correspondence should be addressed.

Axioms 2019, 8(4), 137; https://doi.org/10.3390/axioms8040137

Submission received: 16 November 2019 / Revised: 29 November 2019 / Accepted: 3 December 2019 / Published: 5 December 2019

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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Abstract

Applying Babenko’s approach, we construct solutions for the generalized Abel’s integral equations of the second kind with variable coefficients on R and

R^{n}

, and show their convergence and stability in the spaces of Lebesgue integrable functions, with several illustrative examples.

Keywords:

Riemann–Liouville fractional integral; Mittag–Leffler function; Babenko’s approach; generalized Abel’s integral equation

MSC:

45E10; 26A33

1. Introduction

In 1823, Abel studied a physical problem regarding the relationship between kinetic and potential energies for falling bodies and constructed the integral equation [1,2,3,4]

g (x) = \int_{c}^{x} {(x - t)}^{- 1 / 2} u (t) d t, c > 0,

where

g (x)

is given and

u (x)

is unknown. Later on, he worked on a more general integral equation given as

g (x) = \frac{1}{Γ (α)} \int_{0}^{x} {(x - t)}^{α - 1} u (t) d t, 0 < α < 1, a \leq x \leq b,

which is called Abel’s integral equation of the first kind. Abel’s integral equation of the second kind is generally given as

u (x) - \frac{λ}{Γ (α)} \int_{0}^{x} {(x - t)}^{α - 1} u (t) d t = g (x), α > 0

(1)

where

λ

is a constant.

Abel’s integral equations are related to a wide range of physical problems, such as heat transfer [5], nonlinear diffusion [6], the propagation of nonlinear waves [7], and applications in the theory of neutron transport and traffic theory. There are many studies [8,9,10,11,12,13,14] on Abel’s integral equations, including their variants and generalizations [15,16]. In 1930, Tamarkin investigated integrable solutions of Abel’s integral equations under certain conditions by several integral operators [17]. Sumner [18] studied Abel’s integral equations using the convolutional transform. Minerbo and Levy [19] found a numerical solution of Abel’s integral equation by orthogonal polynomials. In 1985, Hatcher [20] worked on a nonlinear Hilbert problem of power type, solved in closed form by representing a sectionally holomorphic function by means of an integral with power kernel, and transformed the problem to one of solving a generalized Abel’s integral equation. Using a modification of Mikusinski operational calculus, Gorenflo and Luchko [21] obtained an explicit solution of the generalized Abel’s integral equation of the second kind, in terms of the Mittag–Leffler function of several variables.

u (x) - \sum_{i = 1}^{m} λ_{i} (I^{α_{i} μ} u) (x) = g (x), α_{i} > 0, m \geq 1, μ > 0, x > 0

where

λ_{i}

is a constant for

i = 1, 2, \dots, m

, and

I^{μ}

is the Riemann–Liouville fractional integral of order

μ \in R^{+}

with initial point zero [22],

(I^{μ} u) (x) = \frac{1}{Γ (μ)} \int_{0}^{x} {(x - t)}^{μ - 1} u (t) d t .

Lubich [10] constructed the numerical solution for the following Abel’s integral equation of the second kind based on fractional powers of linear multistep methods

u (x) = g (x) + \frac{1}{Γ (α)} \int_{0}^{x} {(x - t)}^{α - 1} f (t, u (t)) d t o n R^{n}

where

x \in [0, T]

and

α > 0

. The case

α = 1 / 2

is encountered in a variety of problems in physics and chemistry [23]. Pskhu [24] considered the following generalized Abel’s integral equation with constant coefficients

a_{k}

for

k = 1, 2, \dots, n

\sum_{k = 1}^{n} a_{k} I^{α_{k}} u (x) = g (x),

where

α_{k} \geq 0

and

x \in (0, a)

, and constructed an explicit solution based on the Wright function

ϕ (α, β; z) = \sum_{n = 0}^{\infty} \frac{z^{n}}{n! Γ (α n + β)}, α > - 1, β \in C

and convolution. Li et al. [25,26,27] recently studied Abel’s integral Equation (1) for any arbitrary

α \in R

in the generalized sense based on fractional calculus of distributions, inverse convolutional operators and Babenko’s approach [28]. They obtained several new and interesting results that cannot be realized in the classical sense or by the Laplace transform. Many applied problems from physical science lead to integral equations which can be converted to the form of Abel’s integral equations for analytic or distributional solutions in the case where classical ones do not exist [15,27].

Letting

α_{1} > α_{2} > \dots > α_{n} > 0

and

a > 0

, we consider the generalized Abel’s integral equation of the second kind with variable coefficients

u (x) - \sum_{k = 1}^{n} a_{k} (x) I^{α_{k}} u (x) = g (x),

(2)

where

x \in (0, a)

a_{i} (x)

is Lebesgue integrable and bounded on

(0, a)

for

i = 1, 2, \dots, n

g (x)

is a given function in

L (0, a)

and

u (x)

is the unknown function. Clearly, Equation (2) turns to be

u (x) - a_{1} I^{α_{1}} u (x) = g (x)

(3)

n = 1

and

a_{1} (x) = a_{1}

(constant). Equation (3) is the classical Abel’s integral equation of the second kind, with the solution given by Hille and Tamarkin [29]

u (x) = g (x) + a_{1} \int_{0}^{x} {(x - t)}^{α_{1} - 1} E_{α_{1}, α_{1}} (a_{1} {(x - t)}^{α_{1}}) g (t) d t,

where

E_{α, β} (z) = \sum_{n = 0}^{\infty} \frac{z^{n}}{Γ (α n + β)}, α, β > 0

is the Mittag–Leffler function.

Following a similar approach, we also establish a convergent and stable solution for the generalized Abel’s integral equation on

R^{n}

with variable coefficients

u (x) - a_{1} (x) I_{1}^{α_{1}} a_{2} (x) I_{2}^{α_{2}} \dots a_{n} (x) I_{n}^{α_{n}} u (x) = g (x),

where

x = (x_{1}, x_{2}, \dots, x_{n})

and

I_{k}^{α}

is the partial Riemann–Liouville fractional integral of order

α \in R^{+}

with respect to

x_{k}

, with initial point 0,

(I_{k}^{α} u) (x) = \frac{1}{Γ (α)} \int_{0}^{x_{k}} {(x_{k} - t)}^{α - 1} u (x_{1}, \dots, x_{k - 1}, t, x_{k + 1}, \dots, x_{n}) d t

where

k = 1, 2, \dots, n

2. The Main Results

Theorem 1.

Let

x \in (0, a)

a_{i} (x)

be Lebesgue integrable and bounded on

(0, a)

for

i = 1, 2, \dots, n

, and

g (x)

be a given function in

L (0, a)

. Then the generalized Abel’s integral equation of the second kind with variable coefficients

u (x) - \sum_{k = 1}^{n} a_{k} (x) I^{α_{k}} u (x) = g (x)

has the following convergent and stable solution in

L (0, a)

u (x) = \sum_{m = 0}^{\infty} {(\sum_{k = 1}^{n} a_{k} (x) I^{α_{k}})}^{m} g (x),

where

α_{1} > α_{2} > \dots > α_{n} > 0

Proof.

Clearly,

u (x) - \sum_{k = 1}^{n} a_{k} (x) I^{α_{k}} u (x) = (1 - \sum_{k = 1}^{n} a_{k} (x) I^{α_{k}}) u (x) = g (x)

which implies, by Babenko’s approach (treating the operator like a variable), that

\begin{matrix} u (x) & = & \frac{1}{1 - \sum_{k = 1}^{n} a_{k} (x) I^{α_{k}}} g (x) = \sum_{m = 0}^{\infty} {(\sum_{k = 1}^{n} a_{k} (x) I^{α_{k}})}^{m} g (x) \\ = & \sum_{m = 0}^{\infty} \sum_{m_{1} + m_{2} + \dots + m_{n} = m} (\binom{m!}{m_{1}!, m_{2}!, \dots, m_{n}!}) {(a_{1} (x) I^{α_{1}})}^{m_{1}} \dots {(a_{n} (x) I^{α_{n}})}^{m_{n}} g (x) . \end{matrix}

Let

∥f∥

be the usual norm of

f \in L (0, a)

, given by

∥f∥ = \int_{0}^{a} | f (x) | d x < \infty .

Then, we have from [30]

∥I^{α_{i}} g∥ = ∥Φ_{α_{i}} * g∥ \leq ∥Φ_{α_{i}}∥ ∥g∥

where

Φ_{α_{i}} = \frac{x_{+}^{α_{i} - 1}}{Γ (α_{i})} .

This implies that

∥I^{α_{i}}∥ \leq ∥Φ_{α_{i}}∥ = \frac{1}{Γ (α_{i})} \int_{0}^{a} x^{α_{i} - 1} = \frac{a^{α_{i}}}{Γ (α_{i} + 1)} .

Since

a_{i} (x)

is bounded over

(0, a)

, there exists

M > 0

such that

sup_{x \in (0, a)} | a_{i} (x) | \leq M

for all

i = 1, 2, \dots, n

. Therefore,

\begin{matrix} ∥u∥ & \leq & \sum_{m = 0}^{\infty} M^{m} \sum_{m_{1} + m_{2} + \dots + m_{n} = m} (\binom{m!}{m_{1}!, m_{2}!, \dots, m_{n}!}) \cdot \\ ∥I^{m_{1} α_{1}}∥ ∥I^{m_{2} α_{2}}∥ \dots ∥I^{m_{n} α_{n}}∥ ∥g∥ \\ \leq & \sum_{m = 0}^{\infty} M^{m} \sum_{m_{1} + m_{2} + \dots + m_{n} = m} (\binom{m!}{m_{1}!, m_{2}!, \dots, m_{n}!}) \cdot \\ \frac{a^{m_{1} α_{1} + \dots + m_{n} α_{n}}}{Γ (m_{1} α_{1} + 1) \dots Γ (m_{n} α_{n} + 1)} ∥g∥ . \end{matrix}

Let

A = max {a, 1} .

Then,

a^{m_{1} α_{1} + \dots + m_{n} α_{n}} \leq A^{m_{1} α_{1} + \dots + m_{n} α_{n}} \leq A^{α_{1} m}

α_{1} > α_{2} > \dots > α_{n} > 0

. On the other hand,

Γ (m_{1} α_{1} + 1) \dots Γ (m_{n} α_{n} + 1) \geq Γ (m_{1} α_{n} + 1) \dots Γ (m_{n} α_{n} + 1) \geq {(\frac{1}{2})}^{n - 1} Γ (α_{n} \frac{m}{n} + 1),

since there exists

m_{i} \geq m / n

for some i by noting that

m_{1} + m_{2} + \dots + m_{n} = m

, and the factor

Γ (m_{j} α_{n} + 1) \geq 1 / 2

for

j \neq i

. Hence,

\frac{1}{Γ (m_{1} α_{1} + 1) \dots Γ (m_{n} α_{n} + 1)} \leq \frac{2^{n - 1}}{Γ (α_{n} \frac{m}{n} + 1)},

and

\begin{matrix} ∥u∥ & \leq & 2^{n - 1} ∥g∥ \sum_{m = 0}^{\infty} \frac{M^{m} n^{m} A^{α_{1} m}}{Γ (α_{n} \frac{m}{n} + 1)} = 2^{n - 1} ∥g∥ \sum_{m = 0}^{\infty} \frac{{(M n A^{α_{1}})}^{m}}{Γ (α_{n} \frac{m}{n} + 1)} \\ = & 2^{n - 1} ∥g∥ E_{α_{n} / n, 1} (M n A^{α_{1}}) < \infty \end{matrix}

by using

\sum_{m_{1} + m_{2} + \dots + m_{n} = m} (\binom{m!}{m_{1}!, m_{2}!, \dots, m_{n}!}) = n^{m} .

Furthermore, the solution

u (x) = \sum_{m = 0}^{\infty} {(\sum_{k = 1}^{n} a_{k} (x) I^{α_{k}})}^{m} g (x)

is stable from the last inequality. This completes the proof of Theorem 1. □

3. Illustrative Examples

Let

α

and

β

be arbitrary real numbers. Then it follows from [31]

Φ_{α} * Φ_{β} = Φ_{α + β} .

Example 1.

Assume

α > 0

. Then Abel’s integral equation with a variable coefficient

u (x) - x^{α} I^{2.5} u (x) = x, x \in (0, a)

has the following stable solution

u (x) = x + \sum_{m = 1}^{\infty} \frac{Γ (α + 4.5) Γ (2 α + 7) \dots Γ (m α + 4.5 + (m - 1) 2.5)}{Γ (4.5) Γ (α + 7) \dots Γ ((m - 1) α + 4.5 + (m - 1) 2.5)} Φ_{m α + 4.5 + (m - 1) 2.5} (x)

L (0, a)

Indeed,

u (x) = x + \sum_{m = 1}^{\infty} {(x^{α} I^{2.5})}^{m} \cdot x = x + \sum_{m = 1}^{\infty} {(x^{α} Φ_{2.5})}^{m} * Φ_{2} .

Clearly,

\begin{matrix} x^{α} Φ_{2.5} * Φ_{2} = x^{α} Φ_{4.5} = \frac{x^{α + 3.5}}{Γ (4.5)} = \frac{Γ (α + 4.5)}{Γ (4.5)} Φ_{α + 4.5}, \\ (x^{α} Φ_{2.5}) * \frac{Γ (α + 4.5)}{Γ (4.5)} Φ_{α + 4.5} = \frac{Γ (α + 4.5)}{Γ (4.5)} x^{α} Φ_{α + 7} = \frac{Γ (α + 4.5)}{Γ (4.5)} \frac{x^{2 α + 6}}{Γ (α + 7)} \\ = \frac{Γ (α + 4.5)}{Γ (4.5)} \frac{Γ (2 α + 7)}{Γ (α + 7)} Φ_{2 α + 7}, \\ \dots, \\ {(x^{α} Φ_{2.5})}^{m} * Φ_{2} = \frac{Γ (α + 4.5) Γ (2 α + 7) \dots Γ (m α + 4.5 + (m - 1) 2.5)}{Γ (4.5) Γ (α + 7) \dots Γ ((m - 1) α + 4.5 + (m - 1) 2.5)} \\ Φ_{m α + 4.5 + (m - 1) 2.5} \end{matrix}

where

m \geq 1

Example 2.

Let

a > 0

. Then Abel’s integral equation

u (x) - x I^{0.5} u (x) - x^{0.5} I u (x) = x^{- 0.5}, x \in (0, a)

has the following stable solution

u (x) = x^{- 0.5} + \sqrt{π} \sum_{m = 1}^{\infty} \sum_{k = 0}^{m} C_{k} B_{m, k} Φ_{2 + 1.5 (m - 1)} (x)

L (0, a)

, where

C_{k} = \{\begin{matrix} 1 & i f k = 0, \\ \frac{Γ (2) Γ (3.5) \dots Γ (2 + 1.5 (k - 1))}{Γ (1.5) Γ (3) \dots Γ (1.5 + 1.5 (k - 1))} & i f k \geq 1 \end{matrix}

and

B_{m, k} = \{\begin{matrix} 1 & i f k = m, \\ \frac{Γ (2 + 1.5 k) Γ (2 + 1.5 (k + 1)) \dots Γ (2 + 1.5 (m - 1))}{Γ (1 + 1.5 k) Γ (1 + 1.5 (k + 1)) \dots Γ (1 + 1.5 (m - 1))} & i f k < m . \end{matrix}

Indeed,

\begin{matrix} u (x) & = & x^{- 0.5} + \sum_{m = 1}^{\infty} {(x I^{0.5} + x^{0.5} I)}^{m} \cdot x^{- 0.5} \\ = & x^{- 0.5} + \sqrt{π} \sum_{m = 1}^{\infty} \sum_{k = 0}^{m} (\binom{m}{k}) {(x Φ_{0.5})}^{m - k} * {(x^{0.5} Φ_{1})}^{k} * Φ_{0.5} . \end{matrix}

Clearly,

\begin{matrix} (x^{0.5} Φ_{1}) * Φ_{0.5} = x^{0.5} Φ_{1.5} = \frac{x}{Γ (1.5)} = \frac{Γ (2)}{Γ (1.5)} Φ_{2}, \\ {(x^{0.5} Φ_{1})}^{2} * Φ_{0.5} = (x^{0.5} Φ_{1}) * \frac{Γ (2)}{Γ (1.5)} Φ_{2} = \frac{Γ (2)}{Γ (1.5)} x^{0.5} Φ_{3} \\ = \frac{Γ (2)}{Γ (1.5)} \frac{x^{2.5}}{Γ (3)} = \frac{Γ (2) Γ (3.5)}{Γ (1.5) Γ (3)} Φ_{3.5}, \\ \dots, \\ {(x^{0.5} Φ_{1})}^{k} * Φ_{0.5} = \frac{Γ (2) Γ (3.5) \dots Γ (2 + 1.5 (k - 1))}{Γ (1.5) Γ (3) \dots Γ (1.5 + 1.5 (k - 1))} Φ_{0.5 + 1.5 k} = C_{k} Φ_{0.5 + 1.5 k} \end{matrix}

where

C_{k}

is defined as above. Furthermore,

\begin{matrix} (x Φ_{0.5}) * Φ_{0.5 + 1.5 k} = x Φ_{1 + 1.5 k} = \frac{x^{1 + 1.5 k}}{Γ (1 + 1.5 k)} = \frac{Γ (2 + 1.5 k)}{Γ (1 + 1.5 k)} Φ_{2 + 1.5 k}, \\ {(x Φ_{0.5})}^{2} * Φ_{0.5 + 1.5 k} = \frac{Γ (2 + 1.5 k)}{Γ (1 + 1.5 k)} x Φ_{2.5 + 1.5 k} = \frac{Γ (2 + 1.5 k)}{Γ (1 + 1.5 k)} x Φ_{1 + 1.5 (k + 1)} \\ = \frac{Γ (2 + 1.5 k) Γ (2 + 1.5 (k + 1))}{Γ (1 + 1.5 k) Γ (1 + 1.5 (k + 1))} Φ_{2 + 1.5 (k + 1)}, \\ \dots, \\ {(x Φ_{0.5})}^{m - k} * Φ_{0.5 + 1.5 k} = \frac{Γ (2 + 1.5 k) Γ (2 + 1.5 (k + 1)) \dots Γ (2 + 1.5 (m - 1))}{Γ (1 + 1.5 k) Γ (1 + 1.5 (k + 1)) \dots Γ (1 + 1.5 (m - 1))} \cdot \\ Φ_{2 + 1.5 (m - 1)} = B_{m, k} Φ_{2 + 1.5 (m - 1)} \end{matrix}

where

B_{m, k}

is defined above.

Remark 1.

As far as we know, the solution for the generalized Abel’s integral equation with variable coefficients over the interval

(0, a)

is obtained for the first time. However, this approach seems unworkable if the interval is unbounded, as the Riemann–Liouville fractional integral operator is therefore unbounded. In the proof and computations of the above examples, we should point out that the convolution operations are prior to functional multiplications, according to our approach.

Assuming that

ω_{i} > 0

for all

i = 1, 2, \dots, n

, and

Ω = (0, ω_{1}) \times (0, ω_{2}) \times \dots \times (0, ω_{n})

, we can derive the following theorem by a similar procedure.

Theorem 2.

Let

α_{k} \geq 0

for

k = 1, 2, \dots, n

and there is at least one

α_{i} > 0

for some

1 \leq i \leq n

. Then the generalized Abel’s integral equation of the second kind with variable coefficients on

R^{n}

for a given function

g \in L (Ω)

u (x) - a_{1} (x) I_{1}^{α_{1}} a_{2} (x) I_{2}^{α_{2}} \dots a_{n} (x) I_{n}^{α_{n}} u (x) = g (x)

has the following convergent and stable solution in

L (Ω)

u (x) = \sum_{m = 0}^{\infty} {(a_{1} (x) I_{1}^{α_{1}} a_{2} (x) I_{2}^{α_{2}} \dots a_{n} (x) I_{n}^{α_{n}})}^{m} g (x),

(4)

where

a_{k} (x)

is Lebesgue integrable and bounded on Ω for

k = 1, 2, \dots, n

Proof.

Clearly,

u (x) - a_{1} (x) I_{1}^{α_{1}} \dots a_{n} (x) I_{n}^{α_{n}} u (x) = (1 - a_{1} (x) I_{1}^{α_{1}} \dots a_{n} (x) I_{n}^{α_{n}}) u (x) = g (x),

and

\begin{matrix} u (x) & = & \frac{1}{1 - a_{1} (x) I_{1}^{α_{1}} \dots a_{n} (x) I_{n}^{α_{n}}} g (x) \\ = & \sum_{m = 0}^{\infty} {(a_{1} (x) I_{1}^{α_{1}} a_{2} (x) I_{2}^{α_{2}} \dots a_{n} (x) I_{n}^{α_{n}})}^{m} g (x) . \end{matrix}

It remains to show that the above is convergent and stable in

L (Ω)

. Let

\begin{matrix} W & = & {(a_{1} (x) I_{1}^{α_{1}} \dots a_{n} (x) I_{n}^{α_{n}})}^{m} \\ = & (a_{1} (x) I_{1}^{α_{1}} \dots a_{n} (x) I_{n}^{α_{n}}) \dots (a_{1} (x) I_{1}^{α_{1}} \dots a_{n} (x) I_{n}^{α_{n}}) . \end{matrix}

Since

a_{k} (x)

is bounded on

Ω

for

k = 1, 2, \dots, n

, there exists

M > 0

such that

sup_{x \in Ω} | a_{k} (x) | \leq M .

Let

∥f∥

be the usual norm of

f \in L (Ω)

, given by

∥f∥ = \int_{Ω} | f (x) | d x = \int_{Ω} | f (x_{1}, x_{2}, \dots, x_{n}) | d x_{1} d x_{2} \dots d x_{n} < \infty .

Then, it follows from [30] for

k = 1, 2, \dots, n

∥I_{k}^{α_{k}} g∥ = ∥Φ_{k, α_{k}} * g∥ \leq ∥Φ_{k, α_{k}}∥ ∥g∥

where

Φ_{k, α_{k}} = \frac{{(x_{k})}_{+}^{α_{k} - 1}}{Γ (α_{k})} .

This implies for

α_{k} > 0

that

\begin{matrix} ∥I_{k}^{α_{k}}∥ \leq ∥Φ_{k, α_{k}}∥ = \int_{Ω} \frac{{(x_{k})}_{+}^{α_{k} - 1}}{Γ (α_{k})} d x_{1} d x_{2} \dots d x_{n} \\ = ω_{1} \dots ω_{k - 1} \frac{ω_{k}^{α_{k}}}{Γ (α_{k} + 1)} ω_{k + 1} \dots ω_{n} \leq λ^{n - 1} \frac{ω_{k}^{α_{k}}}{Γ (α_{k} + 1)} \end{matrix}

where

λ = max {ω_{1}, ω_{2}, \dots, ω_{n}} > 0 .

In particular for

α_{k} = 0

∥I_{k}^{0}∥ \leq λ^{n - 1} .

Therefore,

\begin{matrix} ∥W∥ & \leq & M^{n m} ∥I_{1}^{α_{1} m}∥ \dots ∥I_{n}^{α_{n} m}∥ \\ \leq & M^{n m} λ^{n^{2} - n} \frac{ω_{1}^{α_{1} m}}{Γ (α_{1} m + 1)} \dots \frac{ω_{n}^{α_{n} m}}{Γ (α_{n} m + 1)} \\ \leq & M^{n m} λ^{n^{2} - n} S^{n m} \frac{1}{Γ (α_{1} m + 1)} \dots \frac{1}{Γ (α_{n} m + 1)}, \end{matrix}

where

S = max {ω_{1}^{α_{1}}, \dots, ω_{n}^{α_{n}}} .

Without loss of generality, we assume that

α_{1} > 0

. Then,

Γ (α_{1} m + 1) \dots Γ (α_{n} m + 1) \geq \frac{1}{2^{n - 1}} Γ (α_{1} m + 1)

since

Γ (α_{k} m + 1) \geq 1 / 2

for

k = 2, \dots, n

. This infers that

∥u (x)∥ \leq λ^{n^{2} - n} 2^{n - 1} ∥g∥ \sum_{m = 0}^{\infty} \frac{{(M^{n} S^{n})}^{m}}{Γ (α_{1} m + 1)} < + \infty

by the Mittag–Leffler function. Furthermore, the solution

u (x) = \sum_{m = 0}^{\infty} {(a_{1} (x) I_{1}^{α_{1}} a_{2} (x) I_{2}^{α_{2}} \dots a_{n} (x) I_{n}^{α_{n}})}^{m} g (x)

is stable from the last inequality. This completes the proof of Theorem 2. □

In particular, let

g (x) = ϕ_{1} (x_{1}) \dots ϕ_{n} (x_{n}) \in L (Ω)

. Then

u (x) - a_{1} (x_{1}) I_{1}^{α_{1}} a_{2} (x_{2}) I_{2}^{α_{2}} \dots a_{n} (x_{n}) I_{n}^{α_{n}} u (x) = ϕ_{1} (x_{1}) \dots ϕ_{n} (x_{n})

has the following convergent and stable solution

u (x) = \sum_{m = 0}^{\infty} {(a_{1} (x_{1}) I_{1}^{α_{1}})}^{m} ϕ_{1} (x_{1}) \dots {(a_{n} (x_{n}) I_{n}^{α_{n}})}^{m} ϕ_{n} (x_{n})

L (Ω)

4. Conclusions

We establish the convergent and stable solutions for the following generalized Abel’s integral equations of the second kind with variable coefficients

\begin{matrix} u (x) - \sum_{k = 1}^{n} a_{k} (x) I^{α_{k}} u (x) = g (x), x \in (0, a) \subset R \\ u (x) - a_{1} (x) I_{1}^{α_{1}} a_{2} (x) I_{2}^{α_{2}} \dots a_{n} (x) I_{n}^{α_{n}} u (x) = g (x), x \in Ω \subset R^{n} \end{matrix}

in the spaces of Lebesgue integrable functions, and provide applicable examples based on convolutions and gamma functions.

Author Contributions

The order of the author list reflects contributions to the paper.

Funding

This work is partially supported by NSERC (Canada 2019-03907).

Conflicts of Interest

The authors declare no conflict of interest.

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Li, C.; Plowman, H. Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients. Axioms 2019, 8, 137. https://doi.org/10.3390/axioms8040137

AMA Style

Li C, Plowman H. Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients. Axioms. 2019; 8(4):137. https://doi.org/10.3390/axioms8040137

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Li, Chenkuan, and Hunter Plowman. 2019. "Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients" Axioms 8, no. 4: 137. https://doi.org/10.3390/axioms8040137

APA Style

Li, C., & Plowman, H. (2019). Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients. Axioms, 8(4), 137. https://doi.org/10.3390/axioms8040137

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Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients

Abstract

1. Introduction

2. The Main Results

3. Illustrative Examples

4. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

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