MDPI - Publisher of Open Access Journals

17 pages, 900 KiB

Open AccessArticle

On Hermite–Hadamard–Fejér-Type Inequalities for η-Convex Functions via Quantum Calculus

by Nuttapong Arunrat, Kamsing Nonlaopon and Hüseyin Budak

Mathematics 2023, 11(15), 3387; https://doi.org/10.3390/math11153387 - 2 Aug 2023

Cited by 1 | Viewed by 1115

In this paper, we use

q_{a}

- and

q^{b}

-integrals to establish some quantum Hermite–Hadamard–Fejér-type inequalities for

η

-convex functions. By taking

q \to 1

, our results reduce to classical results on Hermite–Hadamard–Fejér-type inequalities for

η

-convex functions. Moreover, we [...] Read more.

In this paper, we use

q_{a}

- and

q^{b}

-integrals to establish some quantum Hermite–Hadamard–Fejér-type inequalities for

η

-convex functions. By taking

q \to 1

, our results reduce to classical results on Hermite–Hadamard–Fejér-type inequalities for

η

-convex functions. Moreover, we give some examples for quantum Hermite–Hadamard–Fejér-type inequalities for

η

-convex functions. Some results presented here for

η

-convex functions provide extensions of others given in earlier works for convex and

η

-convex functions. Full article

► Show Figures

Figure 1

14 pages, 787 KiB

Open AccessArticle

Generalized Solutions of Ordinary Differential Equations Related to the Chebyshev Polynomial of the Second Kind

by Waritsara Thongthai, Kamsing Nonlaopon, Somsak Orankitjaroen and Chenkuan Li

Mathematics 2023, 11(7), 1725; https://doi.org/10.3390/math11071725 - 4 Apr 2023

Cited by 3 | Viewed by 1710

Abstract

In this work, we employed the Laplace transform of right-sided distributions in conjunction with the power series method to obtain distributional solutions to the modified Bessel equation and its related equation, whose coefficients contain the parameters

ν

and

γ

. We demonstrated that [...] Read more.

In this work, we employed the Laplace transform of right-sided distributions in conjunction with the power series method to obtain distributional solutions to the modified Bessel equation and its related equation, whose coefficients contain the parameters

ν

and

γ

. We demonstrated that the solutions can be expressed as finite linear combinations of the Dirac delta function and its derivatives, with the specific form depending on the values of

ν

and

γ

. Full article

11 pages, 733 KiB

Open AccessArticle

Numerical Analysis of the Fractional-Order Belousov–Zhabotinsky System

by Humaira Yasmin, Azzh Saad Alshehry, Asfandyar Khan, Rasool Shah and Kamsing Nonlaopon

Symmetry 2023, 15(4), 834; https://doi.org/10.3390/sym15040834 - 30 Mar 2023

Cited by 6 | Viewed by 1688

Abstract

This paper presents a new approach for finding analytic solutions to the Belousov–Zhabotinsky system by combining the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) with the Elzaki transform. The ADM and HPM are both powerful techniques for solving nonlinear differential [...] Read more.

This paper presents a new approach for finding analytic solutions to the Belousov–Zhabotinsky system by combining the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) with the Elzaki transform. The ADM and HPM are both powerful techniques for solving nonlinear differential equations, and their combination allows for a more efficient and accurate solution. The Elzaki transform, on the other hand, is a mathematical tool that transforms the system into a simpler form, making it easier to solve. The proposed method is applied to the Belousov–Zhabotinsky system, which is a well-known model for studying nonlinear chemical reactions. The results show that the combined method is capable of providing accurate analytic solutions to the system. Furthermore, the method is also able to capture the complex behavior of the system, such as the formation of oscillatory patterns. Overall, the proposed method offers a promising approach for solving complex nonlinear differential equations, such as those encountered in the field of chemical kinetics. The combination of ADM, HPM, and the Elzaki transform allows for a more efficient and accurate solution, which can provide valuable insights into the behavior of nonlinear systems. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

16 pages, 1080 KiB

Open AccessArticle

Comparative Analysis of Advection–Dispersion Equations with Atangana–Baleanu Fractional Derivative

by Azzh Saad Alshehry, Humaira Yasmin, Fazal Ghani, Rasool Shah and Kamsing Nonlaopon

Symmetry 2023, 15(4), 819; https://doi.org/10.3390/sym15040819 - 29 Mar 2023

Cited by 17 | Viewed by 1966

Abstract

In this study, we solve the fractional advection–dispersion equation (FADE) by applying the Laplace transform decomposition method (LTDM) and the variational iteration transform method (VITM). The Atangana–Baleanu (AB) sense is used to describe the fractional derivative. This equation is utilized to determine solute [...] Read more.

In this study, we solve the fractional advection–dispersion equation (FADE) by applying the Laplace transform decomposition method (LTDM) and the variational iteration transform method (VITM). The Atangana–Baleanu (AB) sense is used to describe the fractional derivative. This equation is utilized to determine solute transport in groundwater and soils. The FADE is converted into a system of non-linear algebraic equations whose solution leads to the approximate solution for this equation using the techniques presented. The proposed approximate method’s convergence is examined. The suggested method’s applicability is demonstrated by testing it on several illustrative examples. The series solutions to the specified issues are obtained, and they contain components that converge more quickly to the precise solutions. The actual and estimated results are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed strategy. The innovation of the current work is in the application of an effective method that requires less calculation and achieves a greater level of accuracy. Furthermore, the proposed approaches may be implemented to prove their utility in tackling fractional-order problems in science and engineering. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

13 pages, 950 KiB

Open AccessArticle

Application of the q-Homotopy Analysis Transform Method to Fractional-Order Kolmogorov and Rosenau–Hyman Models within the Atangana–Baleanu Operator

by Humaira Yasmin, Azzh Saad Alshehry, Abdulkafi Mohammed Saeed, Rasool Shah and Kamsing Nonlaopon

Symmetry 2023, 15(3), 671; https://doi.org/10.3390/sym15030671 - 7 Mar 2023

Cited by 11 | Viewed by 2441

Abstract

The q-homotopy analysis transform method (q-HATM) is a powerful tool for solving differential equations. In this study, we apply the q-HATM to compute the numerical solution of the fractional-order Kolmogorov and Rosenau–Hyman models. Fractional-order models are widely used in physics, engineering, and other [...] Read more.

The q-homotopy analysis transform method (q-HATM) is a powerful tool for solving differential equations. In this study, we apply the q-HATM to compute the numerical solution of the fractional-order Kolmogorov and Rosenau–Hyman models. Fractional-order models are widely used in physics, engineering, and other fields. However, their numerical solutions are difficult to obtain due to the non-linearity and non-locality of the equations. The q-HATM overcomes these challenges by transforming the equations into a series of linear equations that can be solved numerically. The results show that the q-HATM is an effective and accurate method for solving fractional-order models, and it can be used to study a wide range of phenomena in various fields. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

14 pages, 288 KiB

Open AccessArticle

Some Quantum Integral Inequalities for (p, h)-Convex Functions

by Jirawat Kantalo, Fongchan Wannalookkhee, Kamsing Nonlaopon and Hüseyin Budak

Mathematics 2023, 11(5), 1072; https://doi.org/10.3390/math11051072 - 21 Feb 2023

Cited by 3 | Viewed by 1689

Abstract

In this paper, we derive an identity of the q-definite integral of a continuous function f on a finite interval. We then use such identity to prove some new quantum integral inequalities for

(p, h)

-convex function. The results [...] Read more.

In this paper, we derive an identity of the q-definite integral of a continuous function f on a finite interval. We then use such identity to prove some new quantum integral inequalities for

(p, h)

-convex function. The results obtained in this paper generalize previous work in the literature. Full article

20 pages, 1322 KiB

Open AccessArticle

Analysis and Numerical Simulation of System of Fractional Partial Differential Equations with Non-Singular Kernel Operators

by Meshari Alesemi, Jameelah S. Al Shahrani, Naveed Iqbal, Rasool Shah and Kamsing Nonlaopon

Symmetry 2023, 15(1), 233; https://doi.org/10.3390/sym15010233 - 13 Jan 2023

Cited by 6 | Viewed by 2625

Abstract

The exact solution to fractional-order partial differential equations is usually quite difficult to achieve. Semi-analytical or numerical methods are thought to be suitable options for dealing with such complex problems. To elaborate on this concept, we used the decomposition method along with natural [...] Read more.

The exact solution to fractional-order partial differential equations is usually quite difficult to achieve. Semi-analytical or numerical methods are thought to be suitable options for dealing with such complex problems. To elaborate on this concept, we used the decomposition method along with natural transformation to discover the solution to a system of fractional-order partial differential equations. Using certain examples, the efficacy of the proposed technique is demonstrated. The exact and approximate solutions were shown to be in close contact in the graphical representation of the obtained results. We also examine whether the proposed method can achieve a quick convergence with a minimal number of calculations. The present approaches are also used to calculate solutions in various fractional orders. It has been proven that fractional-order solutions converge to integer-order solutions to problems. The current technique can be modified for various fractional-order problems due to its simple and straightforward implementation. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

16 pages, 900 KiB

Open AccessArticle

Investigation of Fractional Nonlinear Regularized Long-Wave Models via Novel Techniques

by Muhammad Naeem, Humaira Yasmin, Rasool Shah, Nehad Ali Shah and Kamsing Nonlaopon

Symmetry 2023, 15(1), 220; https://doi.org/10.3390/sym15010220 - 12 Jan 2023

Cited by 21 | Viewed by 5245

Abstract

The main goal of the current work is to develop numerical approaches that use the Yang transform, the homotopy perturbation method (HPM), and the Adomian decomposition method to analyze the fractional model of the regularized long-wave equation. The shallow-water waves and ion-acoustic waves [...] Read more.

The main goal of the current work is to develop numerical approaches that use the Yang transform, the homotopy perturbation method (HPM), and the Adomian decomposition method to analyze the fractional model of the regularized long-wave equation. The shallow-water waves and ion-acoustic waves in plasma are both explained by the regularized long-wave equation. The first method combines the Yang transform with the homotopy perturbation method and He’s polynomials. In contrast, the second method combines the Yang transform with the Adomian polynomials and the decomposition method. The Caputo sense is applied to the fractional derivatives. The strategy’s effectiveness is shown by providing a variety of fractional and integer-order graphs and tables. To confirm the validity of each result, the technique was substituted into the equation. The described methods can be used to find the solutions to these kinds of equations as infinite series, and when these series are in closed form, they give the precise solution. The results support the claim that this approach is simple, strong, and efficient for obtaining exact solutions for nonlinear fractional differential equations. The method is a strong contender to contribute to the existing literature. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

12 pages, 1260 KiB

Open AccessArticle

Laplace Residual Power Series Method for Solving Three-Dimensional Fractional Helmholtz Equations

by Wedad Albalawi, Rasool Shah, Kamsing Nonlaopon, Lamiaa S. El-Sherif and Samir A. El-Tantawy

Symmetry 2023, 15(1), 194; https://doi.org/10.3390/sym15010194 - 9 Jan 2023

Cited by 14 | Viewed by 2534

Abstract

In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First, we introduce a novel method that combines the [...] Read more.

In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First, we introduce a novel method that combines the Laplace transform tool and the residual power series approach. We specifically give the specifics of how to apply the suggested approach to solve time-fractional nonlinear equations. Second, we use the FHE to evaluate the method’s efficacy and validity. Using 2D and 3D plots of the solutions, the derived and precise solutions are compared, confirming the suggested method’s improved accuracy. The results for nonfractional approximate and accurate solutions, as well as fractional approximation solutions for various fractional orders, are indicated in the tables. The relationship between the derived solutions and the actual solutions to each problem is examined, showing that the solution converges to the actual solution as the number of terms in the series solution of the problems increases. Two examples are shown to demonstrate the effectiveness of the suggested approach in solving various categories of fractional partial differential equations. It is evident from the estimated values that the procedure is precise and simple and that it can therefore be further extended to linear and nonlinear issues. Full article

(This article belongs to the Section Mathematics)

► Show Figures

Figure 1

17 pages, 801 KiB

Open AccessArticle

Investigation of the Time-Fractional Generalized Burgers–Fisher Equation via Novel Techniques

by Badriah M. Alotaibi, Rasool Shah, Kamsing Nonlaopon, Sherif. M. E. Ismaeel and Samir A. El-Tantawy

Symmetry 2023, 15(1), 108; https://doi.org/10.3390/sym15010108 - 30 Dec 2022

Cited by 4 | Viewed by 1941

Abstract

Numerous applied mathematics and physical applications, such as the simulation of financial mathematics, gas dynamics, nonlinear phenomena in plasma physics, fluid mechanics, and ocean engineering, utilize the time-fractional generalized Burgers–Fisher equation (TF-GBFE). This equation describes the concept of dissipation and illustrates how reaction [...] Read more.

Numerous applied mathematics and physical applications, such as the simulation of financial mathematics, gas dynamics, nonlinear phenomena in plasma physics, fluid mechanics, and ocean engineering, utilize the time-fractional generalized Burgers–Fisher equation (TF-GBFE). This equation describes the concept of dissipation and illustrates how reaction systems can be coordinated with advection. To examine and analyze the present evolution equation (TF-GBFE), the modified forms of the Adomian decomposition method (ADM) and homotopy perturbation method (HPM) with Yang transform are utilized. When the results are achieved, they are connected to exact solutions of the

σ = 1

order and even for different values of

σ

to verify the technique’s validity. The results are represented as two- and three-dimensional graphs. Additionally, the study of the precise and suggested technique solutions shows that the suggested techniques are very accurate. Full article

► Show Figures

Figure 1

12 pages, 294 KiB

Open AccessArticle

Certain Solutions of Abel’s Integral Equations on Distribution Spaces via Distributional G_α-Transform

by Supaknaree Sattaso, Kamsing Nonlaopon, Hwajoon Kim and Shrideh Al-Omari

Symmetry 2023, 15(1), 53; https://doi.org/10.3390/sym15010053 - 25 Dec 2022

Cited by 5 | Viewed by 1906

Abstract

Abel’s integral equation is an efficient singular integral equation that plays an important role in diverse fields of science. This paper aims to investigate Abel’s integral equation and its solution using

G_{α}

-transform, which is a symmetric relation between Laplace and Sumudu [...] Read more.

Abel’s integral equation is an efficient singular integral equation that plays an important role in diverse fields of science. This paper aims to investigate Abel’s integral equation and its solution using

G_{α}

-transform, which is a symmetric relation between Laplace and Sumudu transforms.

G_{α}

-transform, as defined via distribution space, is employed to establish a solution to Abel’s integral equation, interpreted in the sense of distributions. As an application to the given theory, certain examples are given to demonstrate the efficiency and suitability of using the

G_{α}

-transform method in solving integral equations. Full article

17 pages, 330 KiB

Open AccessArticle

Fejér-Type Midpoint and Trapezoidal Inequalities for the Operator ω₁,ω₂-Preinvex Functions

by Sikander Mehmood, Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Eman Al-Sarairah, Fiza Zafar and Kamsing Nonlaopon

Axioms 2023, 12(1), 16; https://doi.org/10.3390/axioms12010016 - 24 Dec 2022

Cited by 1 | Viewed by 1581

Abstract

In this work, we obtain some new integral inequalities of the Hermite–Hadamard–Fejér type for operator

(ω_{1}, ω_{2})

-preinvex functions. The bounds for both left-hand and right-hand sides of the integral inequality are established for operator

(ω_{1}, ω_{2})

[...] Read more.

In this work, we obtain some new integral inequalities of the Hermite–Hadamard–Fejér type for operator

(ω_{1}, ω_{2})

-preinvex functions. The bounds for both left-hand and right-hand sides of the integral inequality are established for operator

(ω_{1}, ω_{2})

-preinvex functions of the positive self-adjoint operator in the complex Hilbert spaces. We give the special cases to our results; thus, the established results are generalizations of earlier work. In the last section, we give applications for synchronous (asynchronous) functions. Full article

(This article belongs to the Section Mathematical Analysis)

15 pages, 313 KiB

Open AccessArticle

Dynamic Inequalities of Two-Dimensional Hardy Type via Alpha-Conformable Derivatives on Time Scales

by Ahmed A. El-Deeb, Alaa A. El-Bary, Jan Awrejcewicz and Kamsing Nonlaopon

Symmetry 2022, 14(12), 2674; https://doi.org/10.3390/sym14122674 - 17 Dec 2022

Viewed by 1442

Abstract

We established some new

α

-conformable dynamic inequalities of Hardy–Knopp type. Some new generalizations of dynamic inequalities of

α

-conformable Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy’s inequality for several functions of

α

-conformable calculus. Our [...] Read more.

We established some new

α

-conformable dynamic inequalities of Hardy–Knopp type. Some new generalizations of dynamic inequalities of

α

-conformable Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy’s inequality for several functions of

α

-conformable calculus. Our results are proved by using two-dimensional dynamic Jensen’s inequality and Fubini’s theorem on time scales. When

α = 1

, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we derived Hardy’s inequality for

T = R, T = Z

and

T = h Z

. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. Full article

(This article belongs to the Section Mathematics)

18 pages, 404 KiB

Open AccessArticle

Modified Inequalities on Center-Radius Order Interval-Valued Functions Pertaining to Riemann–Liouville Fractional Integrals

by Soubhagya Kumar Sahoo, Eman Al-Sarairah, Pshtiwan Othman Mohammed, Muhammad Tariq and Kamsing Nonlaopon

Axioms 2022, 11(12), 732; https://doi.org/10.3390/axioms11120732 - 15 Dec 2022

Cited by 7 | Viewed by 1922

Abstract

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, we address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued fractional inequalities for a new class of preinvexity, i.e.,

cr

[...] Read more.

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, we address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued fractional inequalities for a new class of preinvexity, i.e.,

cr

-h-preinvexity, are estimated. The fractional operator is used for the first time to prove such inequalities involving center–radius-ordered functions. Some numerical examples are also provided to validate the presented inequalities. Full article

► Show Figures

Figure 1

9 pages, 272 KiB

Open AccessArticle

On q-Hermite-Hadamard Inequalities via q − h-Integrals

by Yonghong Liu, Ghulam Farid, Dina Abuzaid and Kamsing Nonlaopon

Symmetry 2022, 14(12), 2648; https://doi.org/10.3390/sym14122648 - 15 Dec 2022

Cited by 5 | Viewed by 1654

Abstract

This paper aims to find Hermite–Hadamard-type inequalities for a generalized notion of integrals called

q - h

-integrals. Inequalities for q-integrals can be deduced by taking

h = 0

and are connected with several known results of q-Hermite–Hadamard inequalities. In addition, [...] Read more.

This paper aims to find Hermite–Hadamard-type inequalities for a generalized notion of integrals called

q - h

-integrals. Inequalities for q-integrals can be deduced by taking

h = 0

and are connected with several known results of q-Hermite–Hadamard inequalities. In addition, we analyzed

q - h

-integrals, q-integrals, and the corresponding inequalities for symmetric functions. Full article

(This article belongs to the Section Mathematics)

Search Results (120)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (120)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI