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Open AccessArticle

The Solutions of Some Riemann–Liouville Fractional Integral Equations

by Karuna Kaewnimit, Fongchan Wannalookkhee, Kamsing Nonlaopon and Somsak Orankitjaroen

Fractal Fract. 2021, 5(4), 154; https://doi.org/10.3390/fractalfract5040154 - 6 Oct 2021

Cited by 17 | Viewed by 5800

Abstract

In this paper, we propose the solutions of nonhomogeneous fractional integral equations of the form [...] Read more.

In this paper, we propose the solutions of nonhomogeneous fractional integral equations of the form

I_{0^{+}}^{3 σ} y (t) + a \cdot I_{0^{+}}^{2 σ} y (t) + b \cdot I_{0^{+}}^{σ} y (t) + c \cdot y (t) = f (t),

where

I_{0^{+}}^{σ}

is the Riemann–Liouville fractional integral of order

σ = 1 / 3, 1, f (t) = t^{n}, t^{n} e^{t}, n \in N \cup {0}, t \in R^{+},

and

a, b, c

are constants, by using the Laplace transform technique. We obtain solutions in the form of Mellin–Ross function and of exponential function. To illustrate our findings, some examples are exhibited. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

11 pages, 498 KB

Open AccessArticle

Application of the Laplace Homotopy Perturbation Method to the Black–Scholes Model Based on a European Put Option with Two Assets

by Din Prathumwan and Kamonchat Trachoo

Mathematics 2019, 7(4), 310; https://doi.org/10.3390/math7040310 - 27 Mar 2019

Cited by 8 | Viewed by 4534

Abstract

In this paper, the Laplace homotopy perturbation method (LHPM) is applied to obtain the approximate solution of Black–Scholes partial differential equations for a European put option with two assets. Different from all other approximation methods, LHPM provides a simple way to get the explicit solution which is represented in the form of a Mellin–Ross function. The numerical examples represent that the solution from the proposed method is easy and effective. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)

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11 pages, 953 KB

Open AccessArticle

Laplace Transform Homotopy Perturbation Method for the Two Dimensional Black Scholes Model with European Call Option

by Kamonchat Trachoo, Wannika Sawangtong and Panumart Sawangtong

Math. Comput. Appl. 2017, 22(1), 23; https://doi.org/10.3390/mca22010023 - 27 Feb 2017

Cited by 12 | Viewed by 6323

Abstract

The Black Scholes model is a well-known and useful mathematical model in ﬁnancial markets. In this paper, the two-dimensional Black Scholes equation with European call option is studied. The explicit solution of this problem is carried out in the form of a Mellin–Ross function by using Laplace transform homotopy perturbation method. The solution example demonstrates that the proposed scheme is effective. Full article

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