Research

25 pages, 7649 KiB

Open AccessArticle

A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method

by Agus Sugandha, Endang Rusyaman, Sukono and Ema Carnia

Mathematics 2023, 11(24), 4887; https://doi.org/10.3390/math11244887 - 06 Dec 2023

Cited by 1 | Viewed by 921

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation’s assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

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14 pages, 328 KiB

Open AccessArticle

A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional Integrodifferential Equations

by Abdelkader Moumen, Abdelaziz Mennouni and Mohamed Bouye

Mathematics 2023, 11(18), 3985; https://doi.org/10.3390/math11183985 - 19 Sep 2023

Cited by 4 | Viewed by 740

Abstract

In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method’s efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

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15 pages, 300 KiB

Open AccessArticle

Sharp Existence of Ground States Solutions for a Class of Elliptic Equations with Mixed Local and Nonlocal Operators and General Nonlinearity

by Tingjian Luo and Qihuan Xie

Mathematics 2023, 11(16), 3464; https://doi.org/10.3390/math11163464 - 10 Aug 2023

Viewed by 686

Abstract

In this paper, we study the existence/non-existence of ground states for the following type of elliptic equations with mixed local and nonlocal operators and general nonlinearity: [...] Read more.

In this paper, we study the existence/non-existence of ground states for the following type of elliptic equations with mixed local and nonlocal operators and general nonlinearity:

{(- ▵)}^{s} u - ▵ u + λ u = f (u), x \in R^{N}

, which is driven by the superposition of Brownian and Lévy processes. By considering a constrained variational problem, under suitable assumptions on f, we manage to establish a sharp existence of the ground state solutions to the equation considered. These results improve the ones in the existing reference. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

16 pages, 356 KiB

Open AccessArticle

On Fractional Langevin Equations with Stieltjes Integral Conditions

by Binlin Zhang, Rafia Majeed and Mehboob Alam

Mathematics 2022, 10(20), 3877; https://doi.org/10.3390/math10203877 - 19 Oct 2022

Cited by 4 | Viewed by 1018

Abstract

In this paper, we focus on the study of the implicit

FDE

involving Stieltjes integral boundary conditions. We first exploit some sufficient conditions to guarantee the existence and uniqueness of solutions for the above problems based on the Banach contraction principle and Schaefer’s [...] Read more.

In this paper, we focus on the study of the implicit

FDE

UHS

GUHS

UHRS

, and

GUHRS

by employing the classical techniques. In the end, the main results are demonstrated by two examples. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

40 pages, 783 KiB

Open AccessArticle

Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρ_k,ϕ_k)-Hilfer Fractional Integro-Differential Equations

by Marisa Kaewsuwan, Rachanee Phuwapathanapun, Weerawat Sudsutad, Jehad Alzabut, Chatthai Thaiprayoon and Jutarat Kongson

Mathematics 2022, 10(20), 3874; https://doi.org/10.3390/math10203874 - 18 Oct 2022

Cited by 3 | Viewed by 1150

Abstract

In this paper, we establish the existence and stability results for the

(ρ_{k}, ϕ_{k})

-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the [...] Read more.

In this paper, we establish the existence and stability results for the

(ρ_{k}, ϕ_{k})

-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the

(ρ_{k}, ϕ_{k})

-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

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15 pages, 299 KiB

Open AccessArticle

Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations Involving the Hadamard Derivatives

by Nemat Nyamoradi, Sotiris K. Ntouyas and Jessada Tariboon

Mathematics 2022, 10(17), 3068; https://doi.org/10.3390/math10173068 - 25 Aug 2022

Viewed by 770

Abstract

In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: [...] Read more.

In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative:

_{H} D^{α} x (t) = A f (t, x (t)) + \sum_{i = 1}^{k} C_{i}_{H} I^{β_{i}} g_{i} (t, x (t)), t \in (1, e),

supplemented with fractional Hadamard boundary conditions:

_{H} D^{ξ} x (1) = 0,_{H} D^{ξ} x (e) = a_{H} D^{\frac{α - ξ - 1}{2}} (_{H} D^{ξ} x (t)) |_{t = δ}, δ \in (1, e),

where

1 < α \leq 2

0 < ξ \leq \frac{1}{2}

a \in (0, \infty)

1 < α - ξ < 2

0 < β_{i} < 1

A, C_{i}

1 \leq i \leq k

, are real constants,

_{H} D^{α}

is the Hadamard fractional derivative of order

α

and

_{H} I^{β_{i}}

is the Hadamard fractional integral of order

β_{i}

. By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

17 pages, 358 KiB

Open AccessArticle

Nonlocal Integro-Multi-Point (k, ψ)-Hilfer Type Fractional Boundary Value Problems

by Sotiris K. Ntouyas, Bashir Ahmad, Jessada Tariboon and Mohammad S. Alhodaly

Mathematics 2022, 10(13), 2357; https://doi.org/10.3390/math10132357 - 05 Jul 2022

Cited by 7 | Viewed by 1028

Abstract

In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving

(k, ψ)

-Hilfer fractional derivative operator of order in

(1, 2],

equipped with nonlocal [...] Read more.

In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving

(k, ψ)

-Hilfer fractional derivative operator of order in

(1, 2],

equipped with nonlocal integral multi-point boundary conditions. For the single-valued case, we rely on fixed point theorems due to Banach and Krasnosel’skiĭ, and Leray–Schauder alternative to establish the desired results. The existence results for the multi-valued problem are obtained by applying the Leray–Schauder nonlinear alternative for multi-valued maps for convex-valued case, while the nonconvex-valued case is studied with the aid of Covit–Nadler’s fixed point theorem for multi-valued contractions. Numerical examples are presented for the illustration of the obtained results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

18 pages, 337 KiB

Open AccessArticle

Integrable Solutions for Gripenberg-Type Equations with m-Product of Fractional Operators and Applications to Initial Value Problems

by Ateq Alsaadi, Mieczysław Cichoń and Mohamed M. A. Metwali

Mathematics 2022, 10(7), 1172; https://doi.org/10.3390/math10071172 - 04 Apr 2022

Cited by 7 | Viewed by 1218

Abstract

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line

R^{+} = [0, \infty)

. We prove the existence of solutions in some weighted spaces of [...] Read more.

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line

R^{+} = [0, \infty)

. We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called

L_{1}^{N}

-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

13 pages, 271 KiB

Open AccessArticle

Global Existence for an Implicit Hybrid Differential Equation of Arbitrary Orders with a Delay

by Ahmed M. A. El-Sayed, Sheren A. Abd El-Salam and Hind H. G. Hashem

Mathematics 2022, 10(6), 967; https://doi.org/10.3390/math10060967 - 17 Mar 2022

Cited by 1 | Viewed by 1016

Abstract

In this paper, we present a qualitative study of an implicit fractional differential equation involving Riemann–Liouville fractional derivative with delay and its corresponding integral equation. Under some sufficient conditions, we establish the global and local existence results for that problem by applying some fixed point theorems. In addition, we have investigated the continuous and integrable solutions for that problem. Moreover, we discuss the continuous dependence of the solution on the delay function and on some data. Finally, further results and particular cases are presented. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)

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