Advancements in Weddle's Formula for Tempered Fractional Integrals: Applications in Numerical Integration

Advancements in Weddle’s Formula for Tempered Fractional Integrals: Applications in Numerical Integration Wali Haider1, Hüseyin Budak2, Mehmet Zeki Sarikaya3, Asia Shehzadi4 1School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, Shaanxi, 710119, China 2Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, Türkiye 3 Department of Mathematics Faculty of Science and Arts, Düzce University Düzce 81620, Türkiye 4 School of Mathematics and Statistics, Central South University, Changsha 410083, China Corresponding author: haiderw[email protected] ORCID IDs: First Author: 0009-0001-7065-2755 Second Author: 0000-0001-8843-955X Third Author: 0000-0002-6165-9242 Fourth Author: 0009-0005-1101-5536 DOI : 10.5281/zenodo.18033675 Abstract Tempered fractional integrals arise naturally in the modeling of anomalous diffusion, viscoelastic phenomena, and other processes in modern physics. In this study, we provides a comprehensive derivation of integral inequalities for differentiable convex functions in the context of tempered fractional integrals. Initially, we formulate significant integral identity involving tempered fractional integrals, which are subsequently used to establish Weddle’s type inequalities for differentiable convex functions. Furthermore, numerical demonstrations and graphical representations to affirm the validity and relevance of these newly derived inequalities within the tempered fractional integral framework are presented. Keywords: Convexity, Weddle’s Formula, Fractional Calculus, Tempered Fractional Integrals, Quadrature Formula 1 Introduction and Motivation Numerical integration has an extensive and diverse past traceable to ancient civilizations. The initial attempts to approximate integrals can be traced back to ancient civilizations, where geometric methods were employed to calculate areas and volumes. Areas under curves were first calculated by Archimedes, who invented the fundamental techniques. Newton and Leibniz’s calculus breakthroughs established the framework for modern numerical approaches, such as M36-1 KOSC-2025 Proceedings Newton’s trapezoidal rule and formalized integration in the 17th century. Gauss significantly contributed to the development of numerical integration by introducing the concept of Gaussian quadrature in the 19th century. Further developments, such as the Monte Carlo methods and Romberg integration, occurred in the 20th century, and the rise of modern computing tools reinforced numerical integration’s position as an essential tool for solving challenging issues in various scientific domains. These techniques compute a weighted sum of function values over the prescribed interval to approximate the integral. Numerous standard methods are present, each with distinctive advantages based on the function’s complexity and required accuracy, including the Gaussian quadrature, Trapezoidal rule, Simpson’s rule, Boole’s rule, Weddle’s rule and Monte Carlo approaches. Consult [ 1 , 4 , 16 ] and references therein for further information on numerical integration and its applications. In mathematics, the study of inequalities has been prevalent since ancient times. However, their formal and systematic studies have been more pronounced since the 17th and 18th centuries. Karl Hermann Amandus Schwarz [6] significantly contributed to this approach in the late 19th century, particularly with the formation of convex-shaped functions. This concept substantially enhanced the field of mathematics and became an essential aspect of optimization, which further affected other fields like economics, engineering, and computer science. At present, convex functions are also highly applicable in solving complicated optimization problems, such as portfolio optimization and control theory operations. For additional details on the development in time and the field of convexity, see [ 22 , 25 ]. Convex functions have been employed for many years to investigate inequalities of the Hermite–Hadamard, Simpson, Hermite–Hadamard-Mercer, Ostrowski, and other types. The Hermite–Hadamard inequality, which is discussed in [ 9 ], has captured the attention of a diverse array of scholars. Dragomir [ 5 ] and Kirmaci[ 14 ] have provided numerous inequalities of the trapezoidal type, which have applications to special means. The prominent numerical integration techniques include Simpson’s formula, Midpoint formula, Trapezoidal formula, Boole’s formula, and Weddle’s formula. These methods are intended to approximate the integral of a function over a specified interval by dividing it into smaller subintervals and computing the area under the curve employing several methods. Here are mention some well-known quadrature formulas: (a) Simpson’s quadrature, which is also known as Simpson’s 1/3rule, can be indicated as: Zυ κ Φ(t)dt ≈υ−κ 6Φ(κ) + 4Φκ+υ 2+ Φ(υ).(1) (b) Simpson’s second formula, or Newton-Cotes quadrature, can be stated in the following manner [4]: Zυ κ Φ(t)dt ≈υ−κ 8Φ(κ) + 3Φ2κ+υ 3+ 3Φκ+ 2υ 3+ Φ(υ).(2) (c) The Maclaurin rule, obtained from the Maclaurin formula (as given in [ 4 ],) is equivalent to the similar dual Simpson’s 3/8formula: Zυ κ Φ(t)dt ≈υ−κ 83Φ5κ+υ 6+ 2Φκ+υ 2+ 3Φκ+ 5υ 6.(3) M36-2 2nd Kocaeli Science Congress, November 19-21, 2025 These formulas apply to any function Φhaving a continuous fourth derivative across the interval [κ, υ]: (1), (2), and (3). (d) The Boole’s formula, stated as: Zυ κ Φ(t)dt ≈υ−κ 90 7[Φ(κ) + Φ(υ)] + 32Φυ+ 3κ 4+ Φ3υ+κ 4+ 12Φκ+υ 2. (4) (e) The Weddle’s formula, stated as: Zυ κ Φ(t)dt ≈1 20 Φ(κ) + Φ(υ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2 +5Φ5υ+κ 6+ Φ2υ+κ 3.(5) The fractional calculus theory expands the regular differentiation and integration notions and applies non-integer values for analyzing various dynamic systems. Tempered fractional integrals, a particular type of fractional calculus, include an exponential decay factor that enhances accuracy in systems with localized interactions and fading memory. By implementing these tempered integrals, researchers achieve more accurate and realistic models for various scientific and engineering applications. Its importance is in using the method to approximate better and describe problem experiences in different fields, including engineering, biosciences, data processing, image enhancement, finance, and other scientific disciplines have extensive applications for tempered fractional integrals. The mathematical preliminaries and definitions outlined here will be used extensively throughout this investigation. Definition 1.1. The definitions of the incomplete gamma function and the λ -incomplete gamma function are outlined in the following way: ⋎(α, ϖ) := ϖ Z 0 tα−1e−tdt, and ⋎λ(α, ϖ) := ϖ Z 0 tα−1e−λtdt. Here, 0<α<∞and λ≥0. The characteristics of the λ-incomplete gamma function are outlined as: Remark 1.1. [20] Under the assumptions α > 0;x, λ ≥0and a < b, we attain i. ⋎λ(υ−κ)(α, 1) = 1 R0 tα−1e−λ(υ−κ)tdt =1 (υ−κ)α⋎λ(α, υ −κ), ii. 1 R0 ⋎λ(υ−κ)(α, ϖ)dϖ =⋎λ(α,υ−κ) (υ−κ)α−⋎λ(α+1,υ−κ) (υ−κ)α+1 . 2nd Kocaeli Science Congress, November 19-21, 2025 M36-3 KOSC-2025 Proceedings Definition 1.2 (See [ 13 , 8 ]).A mapping Φgiven on the interval [ κ, υ ], assume Φ ∈L1 [ κ, υ ], the Riemann-Liouville fractional integral of order α > 0, stated as follows: Jα κ+Φ(ϖ) = 1 Γ(α)Zϖ κ (ϖ−t)α−1Φ(t)dt, ϖ > κ and Jα υ−Φ(ϖ) = 1 Γ(α)Zυ ϖ (t−ϖ)α−1Φ(t)dt, ϖ < υ. Here, Γ(α)is the Gamma function and J0 κ+Φ(ϖ) = J0 υ−Φ(ϖ) = Φ(ϖ). Definition 1.3 (See [ ? , 17 ]).The definitions of the tempered fractional integral operators outlined as: Z(α,λ) κ+Φ(ϖ) = 1 Γ(α)Zϖ κ (ϖ−t)α−1e−λ(ϖ−t)Φ(t)dt, ϖ ∈[κ, υ] and Z(α,λ) υ−Φ(ϖ) = 1 Γ(α)Zυ ϖ (t−ϖ)α−1e−λ(t−ϖ)Φ(t)dt, ϖ ∈[κ, υ]. Here, Φ∈L1[κ, υ], α > 0and λ≥0. Definition 1.2 is immediately obtained if λ = 0 is substituted in Definition 1.3. The following books should be consulted to explore further the diverse cases of tempered fractional integrals [24,19]. The innovative study of Buschman [ 2 ] proposed the concept of fractional integration involving weak singular and exponential kernels, which is the source of tempering fractional calculus, which evolved from the principles of fractional calculus. Fu et al. [ 7 ] obtained a novel class of the multiplicative tempered fractional integral operators. They examined various integral inequalities employing λ -incomplete gamma function and multiplicative tempered fractional integrals. Rehman et al. [ 21 ] have investigated the relationship between Riemann-Louville fractional integral tempered fractional. Also, they identified several new inequalities for convex function in terms of tempered fractional integrals. In [ 20 ], authors revealed a concept for λ -incomplete gamma functions. Based on this particular notion, they acquired several HermiteHadamard inequalities in the framework of tempered fractional integrals. Several HermiteHadamard type inequalities involving tempered fractional integrals for subadditive functions were obtained in [ 12 ]. In the setting of tempered fractional calculus, Kucche et al. [ 15 ] have investigated a Hilfer-type operator. They analyzed its key characteristics, such as compositional properties, mappings in function spaces, and other functional analysis characteristic. Numerous researchers have contributed to advancing tempered fractional integral theories; see [ 3 , 11 , 23 , 10 ] and the references therein. Motivated by previous investigations, we acquire some of Weddle’s formula-type inequalities for the class of functions whose derivatives are convex functions employing tempered fractional integrals. In this work, we examine the significant classes of functions in the setting of tempered fractional integral, particularly convex functions, Hölder, and power mean inequalities. A notable advantage of these inequalities can be turned into Riemann-Liouville fractional Weddle’s type inequalities with λ = 0. When α = 1, the modified inequalities yield classical Weddle’s type inequalities. M36-4 2nd Kocaeli Science Congress, November 19-21, 2025 1.1 Main Contributions The study is organized into four sections, starting with the introduction and preliminaries, which cover basic definitions of fractional calculus and a summary of relevant studies in the field. In Section 1.1, we will establish several results concerning Weddle’s type inequalities for differentiable convex functions. Section 2, we will provide various examples to elucidate these inequalities, highlighting their practical applications and significance. In the last section, we summarize our findings and suggest possibilities for further research. 1.1 Main Contributions Lemma 1.1. Let Φ:[ κ, υ ] → Ris an absolutely continuous function on the interval ( κ, υ )with Φ′∈L1[κ, υ], then the following holds: 1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α) 2⋎λ(α, υ −κ)hZ(α,λ) υ−Φ(κ)+Z(α,λ) κ+Φ(υ)i=(υ−κ)α+1 2⋎λ(α, υ −κ) 6 X i=1 Ii,(6) where                                                        I1=Z1 6 0⋎λ(υ−κ)(α, t)−1 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt, I2=Z1 3 1 6⋎λ(υ−κ)(α, t)−6 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt, I3=Z1 2 1 3⋎λ(υ−κ)(α, t)−7 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt, I4=Z2 3 1 2⋎λ(υ−κ)(α, t)−13 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt, I5=Z5 6 2 3⋎λ(υ−κ)(α, t)−14 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt, I6=Z1 5 6⋎λ(υ−κ)(α, t)−19 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt. Proof. Through the technique of integration by parts, we establish I1=Z1 6 0⋎λ(υ−κ)(α, t)−1 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt =1 υ−κ⋎λ(υ−κ)(α, t)−1 20 ⋎λ(υ−κ)(α, 1)[Φ(tυ + (1 −t)κ) + Φ(tκ + (1 −t)υ)] 1 6 0 −1 υ−κZ1 6 0 tα−1e−λ(υ−κ)t[Φ(tυ +(1−t)κ) + Φ(tκ + (1 −t)υ)]dt =1 υ−κ⋎λ(υ−κ)α, 1 6−1 20 ⋎λ(υ−κ)(α, 1)Φυ+ 5κ 6+ Φκ+ 5υ 6 2nd Kocaeli Science Congress, November 19-21, 2025 M36-5 KOSC-2025 Proceedings +1 υ−κ1 20 ⋎λ(υ−κ)(α, 1)[Φ(κ) + Φ(υ)] −1 υ−κZ1 6 0 tα−1e−λ(υ−κ)t[Φ(tυ + (1 −t)κ) + Φ(tκ + (1 −t)υ)]dt, (7) I2=Z1 3 1 6⋎λ(υ−κ)(α, t)−6 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt =1 υ−κ⋎λ(υ−κ)α, 1 3−6 20 ⋎λ(υ−κ)(α, 1)Φυ+ 2κ 3+ Φκ+ 2υ 3 −1 υ−κ⋎λ(υ−κ)α, 1 6−6 20 ⋎λ(υ−κ)(α, 1)Φυ+ 5κ 6+ Φκ+ 5υ 6 −1 υ−κZ1 3 1 6 tα−1e−λ(υ−κ)t[Φ(tυ +(1−t)κ) + Φ(tκ + (1 −t)υ)]dt, (8) I3=Z1 2 1 3⋎λ(υ−κ)(α, t)−7 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt =2 υ−κ⋎λ(υ−κ)α, 1 2−7 20 ⋎λ(υ−κ)(α, 1)Φκ+υ 2 −1 υ−κ⋎λ(υ−κ)α, 1 3−7 20 ⋎λ(υ−κ)(α, 1)Φυ+ 2κ 3+ Φκ+ 2υ 3 −1 υ−κZ1 2 1 3 tα−1e−λ(υ−κ)t[Φ(tυ +(1−t)κ) + Φ(tκ + (1 −t)υ)]dt, (9) I4=Z2 3 1 2⋎λ(υ−κ)(α, t)−13 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt =1 υ−κ⋎λ(υ−κ)α, 2 3−13 20 ⋎λ(υ−κ)(α, 1)Φ2υ+κ 3+ Φ2κ+υ 3 −2 υ−κ⋎λ(υ−κ)α, 1 2−13 20 ⋎λ(υ−κ)(α, 1)Φκ+υ 2 −1 υ−κZ2 3 1 2 tα−1e−λ(υ−κ)t[Φ(tυ +(1−t)κ) + Φ(tκ + (1 −t)υ)]dt, (10) I5=Z5 6 2 3⋎λ(υ−κ)(α, t)−14 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt =1 υ−κ⋎λ(υ−κ)α, 5 6−14 20 ⋎λ(υ−κ)(α, 1)Φ5υ+κ 6+ Φ5κ+υ 6 −1 υ−κ⋎λ(υ−κ)α, 2 3−14 20 ⋎λ(υ−κ)(α, 1)Φ2υ+κ 3+ Φ2κ+υ 3 −1 υ−κZ5 6 2 3 tα−1e−λ(υ−κ)t[Φ(tυ +(1−t)κ) + Φ(tκ + (1 −t)υ)]dt, (11) and I6=Z1 5 6⋎λ(υ−κ)(α, t)−19 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt =1 υ−κ1 20 ⋎λ(υ−κ)(α, 1)[Φ (κ) + Φ (υ)] −1 υ−κ⋎λ(υ−κ)α, 5 6−19 20 ⋎λ(υ−κ)(α, 1)Φ5υ+κ 6+ Φ5κ+υ 6 M36-6 2nd Kocaeli Science Congress, November 19-21, 2025 1.1 Main Contributions −1 υ−κZ1 5 6 tα−1e−λ(υ−κ)t[Φ(tυ + (1 −t)κ) + Φ(tκ + (1 −t)υ)]dt. (12) By adding equalities (7)-(12), we acquire 6 X i=1 Ii=2⋎λ(α, υ −κ) 20(υ−κ)α+1 ×Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −1 υ−κZ1 0 tα−1e−λ(υ−κ)t[Φ(tυ +(1−t)κ) + Φ(tκ + (1 −t)υ)]dt. (13) With the aid of change of variable x = tυ +(1 −t ) κ and x = tκ +(1 −t ) υ for t∈ [0 , 1] respectively, equality (13) can be rewrited as: 6 X i=1 Ii=2⋎λ(α, υ −κ) 20(υ−κ)α+1 ×Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α) (υ−κ)α+1 hZ(α,λ) υ−Φ(κ)+Z(α,λ) κ+Φ(υ)i.(14) Multiplying both sides of (14) by (υ−κ)α+1 2⋎λ(α,υ−κ), the equality (6) is obtained. Theorem 1.1. Suppose the conditions stated in Lemma 1.1 and the function | Φ ′| on the interval [κ, υ], then, we have the subsequent inequality  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α) 2⋎λ(α, υ −κ)hZ(α,λ) υ−Φ(κ)+Z(α,λ) κ+Φ(υ)i ≤(υ−κ)α+1 2⋎λ(α, υ −κ)(K1(α, λ) + K2(α, λ) + K3(α, λ) + K4(α, λ) + K5(α, λ) + K6(α, λ))|Φ′(κ)|+|Φ′(υ)|, (15) where                                                        K1(α, λ) = Z1 6 0 ⋎λ(υ−κ)(α, t)−1 20 ⋎λ(υ−κ)(α, 1) dt, K2(α, λ) = Z1 3 1 6 ⋎λ(υ−κ)(α, t)−6 20 ⋎λ(υ−κ)(α, 1) dt, K3(α, λ) = Z1 2 1 3 ⋎λ(υ−κ)(α, t)−7 20 ⋎λ(υ−κ)(α, 1) dt, K4(α, λ) = Z2 3 1 2 ⋎λ(υ−κ)(α, t)−13 20 ⋎λ(υ−κ)(α, 1) dt, K5(α, λ) = Z5 6 2 3 ⋎λ(υ−κ)(α, t)−14 20 ⋎λ(υ−κ)(α, 1) dt, K6(α, λ) = Z1 5 6 ⋎λ(υ−κ)(α, t)−19 20 ⋎λ(υ−κ)(α, 1) dt. 2nd Kocaeli Science Congress, November 19-21, 2025 M36-7 KOSC-2025 Proceedings Proof. By empolying the modulus in Lemma 1.1 and considering the convexity of |Φ′|, we gain  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α) 2⋎λ(α, υ −κ)hZ(α,λ) υ−Φ(κ)+Z(α,λ) κ+Φ(υ)i ≤(υ−κ)α+1 2⋎λ(α, υ −κ)"Z1 6 0 ⋎λ(υ−κ)(α, t)−1 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt +Z1 3 1 6 ⋎λ(υ−κ)(α, t)−6 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ +(1−t)κ)−Φ′(tκ + (1 −t)υ)dt +Z1 2 1 3 ⋎λ(υ−κ)(α, t)−7 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ +(1−t)κ)−Φ′(tκ + (1 −t)υ)dt +Z2 3 1 2 ⋎λ(υ−κ)(α, t)−13 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ +(1−t)κ)−Φ′(tκ + (1 −t)υ)dt +Z5 6 2 3 ⋎λ(υ−κ)(α, t)−14 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ +(1−t)κ)−Φ′(tκ + (1 −t)υ)dt +Z1 5 6 ⋎λ(υ−κ)(α, t)−19 20 ⋎λ(υ−κ)(α, 1)Φ′(tυ + (1 −t)κ)−Φ′(tκ + (1 −t)υ)dt#.(16) By using the convexity of |Φ′|, we can deduce  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α) 2⋎λ(α, υ −κ)hZ(α,λ) υ−Φ(κ)+Z(α,λ) κ+Φ(υ)i ≤(υ−κ)α+1 2⋎λ(α, υ −κ) ×"Z1 6 0 ⋎λ(υ−κ)(α, t)−1 20 ⋎λ(υ−κ)(α, 1)t|Φ′(υ)|+(1−t)|Φ′(κ)|+t|Φ′(κ)|+ (1 −t)|Φ′(υ)|dt +Z1 3 1 6 ⋎λ(υ−κ)(α, t)−6 20 ⋎λ(υ−κ)(α, 1)t|Φ′(υ)|+ (1 −t)|Φ′(κ)|+t|Φ′(κ)|+ (1 −t)|Φ′(υ)|dt +Z1 2 1 3 ⋎λ(υ−κ)(α, t)−7 20 ⋎λ(υ−κ)(α, 1)t|Φ′(υ)|+ (1 −t)|Φ′(κ)|+t|Φ′(κ)|+ (1 −t)|Φ′(υ)|dt +Z2 3 1 2 ⋎λ(υ−κ)(α, t)−13 20 ⋎λ(υ−κ)(α, 1)t|Φ′(υ)|+ (1 −t)|Φ′(κ)|+t|Φ′(κ)|+ (1 −t)|Φ′(υ)|dt +Z5 6 2 3 ⋎λ(υ−κ)(α, t)−14 20 ⋎λ(υ−κ)(α, 1)t|Φ′(υ)|+ (1 −t)|Φ′(κ)|+t|Φ′(κ)|+ (1 −t)|Φ′(υ)|dt +Z1 5 6 ⋎λ(υ−κ)(α, t)−19 20 ⋎λ(υ−κ)(α, 1)t|Φ′(υ)|+(1−t)|Φ′(κ)|+t|Φ′(κ)|+ (1 −t)|Φ′(υ)|dt# =(υ−κ)α+1 2⋎λ(α, υ −κ)(K1(α, λ) + K2(α, λ) + K3(α, λ) + K4(α, λ) + K5(α, λ) + K6(α, λ))|Φ′(κ)|+|Φ′(υ)|. Hence, the proof of Theorem 1.1 is established. M36-8 2nd Kocaeli Science Congress, November 19-21, 2025 1.1 Main Contributions Remark 1.2. By taking λ= 0 in Theorem 1.1, we attain  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α+ 1) 2(υ−κ)αJα υ−Φ(κ)+Jα κ+Φ(υ) ≤α(υ−κ) 2(K1(α, 0) + K2(α, 0) + K3(α, 0) + K4(α, 0) + K5(α, 0) + K6(α, 0)), which is identified in [18]. Remark 1.3. Let us consider λ= 0 and α= 1 in Theorem 1.1, then we acquire  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −1 υ−κZυ κ Φ(t)dt ≤13(υ−κ) 450 |Φ′(κ)|+|Φ′(υ)|, which is reported in [26]. Theorem 1.2. Suppose the conditions stated in Lemma 1.1. If the mapping | Φ ′|q, q ≥ 1is convex on the interval [κ, υ], then we have the subsequent inequality  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α) 2⋎λ(α, υ −κ)hZ(α,λ) υ−Φ(κ)+Z(α,λ) κ+Φ(υ)i ≤(υ−κ)α+1 2⋎λ(α, υ −κ)(K1(α, λ))1−1 qK7(α, λ)|Φ′(υ)|q+ (K1(α, λ)−K7(α, λ)) |Φ′(κ)|q1 q +K7(α, λ)|Φ′(κ)|q+ (K1(α, λ)−K7(α, λ)) |Φ′(υ)|q1 q + (K2(α, λ))1−1 qK8(α, λ)|Φ′(υ)|q+ (K2(α, λ)−K8(α, λ)) |Φ′(κ)|q1 q +K8(α, λ)|Φ′(κ)|q+ (K2(α, λ)−K8(α, λ)) |Φ′(υ)|q1 q + (K3(α, λ))1−1 qK9(α, λ)|Φ′(υ)|q+ (K3(α, λ)−K9(α, λ)) |Φ′(κ)|q1 q +K9(α, λ)|Φ′(κ)|q+ (K3(α, λ)−K9(α, λ)) |Φ′(υ)|q1 q + (K4(α, λ))1−1 qK10(α, λ)|Φ′(υ)|q+ (K4(α, λ)−K10(α, λ))|Φ′(κ)|q1 q +K10(α, λ)|Φ′(κ)|q+ (K4(α, λ)−K10(α, λ))|Φ′(υ)|q1 q + (K5(α, λ))1−1 qK11(α, λ)|Φ′(υ)|q+ (K5(α, λ)−K11(α, λ))|Φ′(κ)|q1 q +K11(α, λ)|Φ′(κ)|q+ (K5(α, λ)−K11(α, λ))|Φ′(υ)|q1 q + (K6(α, λ))1−1 qK12(α, λ)|Φ′(υ)|q+ (K6(α, λ)−K12(α, λ))|Φ′(κ)|q1 q +K12(α, λ)|Φ′(κ)|q+ (K6(α, λ)−K12(α, λ))|Φ′(υ)|q1 q,(17) 2nd Kocaeli Science Congress, November 19-21, 2025 M36-9 KOSC-2025 Proceedings =(υ−κ)α+1 2⋎λ(α, υ −κ) Bp 1(α, λ)   |Φ′(υ)|q+ 11|Φ′(κ)|q 72 !1 q + 11|Φ′(υ)|q+|Φ′(κ)|q 72 !1 q   +Bp 2(α, λ)   |Φ′(υ)|q+ 3|Φ′(κ)|q 24 !1 q + 3|Φ′(υ)|q+|Φ′(κ)|q 24 !1 q   +Bp 3(α, λ)   5|Φ′(υ)|q+ 7|Φ′(κ)|q 72 !1 q + 7|Φ′(υ)|q+ 5|Φ′(κ)|q 72 !1 q   +Bp 4(α, λ)   5|Φ′(υ)|q+ 7|Φ′(κ)|q 72 !1 q + 7|Φ′(υ)|q+ 5|Φ′(κ)|q 72 !1 q   +Bp 5(α, λ)   |Φ′(υ)|q+ 3|Φ′(κ)|q 24 !1 q + 3|Φ′(υ)|q+|Φ′(κ)|q 24 !1 q   +Bp 6(α, λ)   |Φ′(υ)|q+ 11|Φ′(κ)|q 72 !1 q + 11|Φ′(υ)|q+|Φ′(κ)|q 72 !1 q   . Therefore, the proof is established. Remark 1.6. Let us consider λ= 0 in Theorem 1.3, we get  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −Γ(α+ 1) 2(υ−κ)αJα υ−Φ(κ)+Jα κ+Φ(υ) ≤α(υ−κ) 2  Bp 1(α, 0) + Bp 6(α, 0)  |Φ′(υ)|q+ 11|Φ′(κ)|q 72 !1 q + 11|Φ′(υ)|q+|Φ′(κ)|q 72 !1 q  +Bp 2(α, 0) + Bp 5(α, 0)  |Φ′(υ)|q+ 3|Φ′(κ)|q 24 !1 q + 3|Φ′(υ)|q+|Φ′(κ)|q 24 !1 q  +Bp 3(α, 0) + Bp 4(α, 0)  |5Φ′(υ)|q+ 7|Φ′(κ)|q 72 !1 q + 7|Φ′(υ)|q+ 5|Φ′(κ)|q 72 !1 q    , which is presented in [18]. Remark 1.7. Let us consider λ= 0 and α= 1 in Theorem 1.3, then we attain  1 20 Φ(κ) + 5Φυ+ 5κ 6+ Φυ+ 2κ 3+ 6Φκ+υ 2+ Φ2υ+κ 3+ 5Φ5υ+κ 6+ Φ(υ) −1 υ−κZυ κ Φ(t)dt ≤    2  3−p−17p20−2p−13p+1 20 7p+ 7 20p p+ 1   1 p ×   |Φ′(υ)|q+ 11|Φ′(κ)|q 72 !1 q + 11|Φ′(υ)|q+|Φ′(κ)|q 72 !1 q   M36-16 2nd Kocaeli Science Congress, November 19-21, 2025 + 21−1 p  15−2p−115 2p+ 2p+215p p+ 1   1 p ×   |Φ′(υ)|q+ 3|Φ′(κ)|q 24 !1 q + 3|Φ′(υ)|q+|Φ′(κ)|q 24 !1 q   +2×3−1/p   20−2p−120 3p+ 3p+220p p+ 1   1 p ×   |Φ′(υ)|q+ 11|Φ′(κ)|q 72 !1 q + 11|Φ′(υ)|q+|Φ′(κ)|q 72 !1 q   , which is identified in [18, Corollary 3]. 2 Numerical Examples This section includes an extensive numerical analysis to validate the effectiveness of recently derived results. The proposed inequalities are significant for approximating the integrals of differentiable convex functions, as demonstrated by several numerical examples. Example 2.1. Consider two differentiable convex functions Φ( t ) = t6 and Φ( t ) = et respectively, in Theorem 1.1 for all t > 0 , λ = 1, κ = 1 and υ = 2, then we observe the subsequent numerical verification for inequality (15) (see Tables 1and 2) and corresponding graphs (see Figures 2,1,4 and 3). (a) Figure 1: Graphically representation of Theorem 1.1, and Example 2.1: (a) indicates 3-D plot, for Φ(t)=t6. When λis fixed and αvaries from 0to 1, computed and plotted with Mathematica. Remark 2.1. As one can easily observe from 2-D Figures 2and 4the left-hand side of (15) in Example 2.1 is always below the right-hand side. We explored Weddle’s inequality behavior about tempered fractional calculus for Theorem 1.1. These graphs might show where singularities or breakpoints arise and how inequality holds under certain conditions. 2nd Kocaeli Science Congress, November 19-21, 2025 M36-17 KOSC-2025 Proceedings (a) 2D plot for α∈[0,0.53) (b) 2D plot for α∈[0.53,0.87) (c) 2D plot for α∈[0.87,0.93) (d) 2D plot for α∈[0.93,1] Figure 2: Graphical representation of Theorem 1.1, and Example 2.1: (a), (b), (c), and (d) 2-D plots, when Φ( t ) = t6 , λ is fixed and α varies from 0to 1, computed and plotted with Mathematica. (a) Figure 3: Graphically representation of Theorem 1.1, and Example 2.1: (a) indicates 3-D plot, for Φ(t)=et. When λis fixed and αvaries from 0to 1, computed and plotted with Mathematica. Remark 2.2. As one can easily observe from 3-D Figures 1and 3also Tables 1and 2the left-hand side of (15) in Example 2.1 is always below the right-hand side. We explored Weddle’s inequality behavior about tempered fractional calculus for Theorem 1.1. M36-18 2nd Kocaeli Science Congress, November 19-21, 2025 2.1 Future Research Directions (a) 2D plot for α∈[0,0.53) (b) 2D plot for α∈[0.53,0.87) (c) 2D plot for α∈[0.87,0.93) (d) 2D plot for α∈[0.93,1] Figure 4: Graphically representation of Theorem 1.1, and Example 2.1: (a), (b), (c), and (d) 2-D plots, when Φ( t ) = et , λ is fixed and α varies from 0to 1, computed and plotted with Mathematica. αLeft Term Right Term 0.1 11.4205 63.8124 0.2 8.9961 79.8606 0.3 6.9974 97.9656 0.4 5.3529 118.4931 0.5 4.0032 141.8627 0.6 2.8992 267.8510 0.7 2.0005 325.7008 0.8 1.2731 392.9232 0.9 0.6893 730.0312 1.0 0.2258 1151.4531 Table 1: Numerical values of inequalities (15) for Φ(t)=t6,κ= 1, and υ= 2. αLeft Term Right Term 0.1 0.3055 3.2574 0.2 0.2413 4.0767 0.3 0.1882 5.0009 0.4 0.1443 6.0487 0.5 0.1081 7.2417 0.6 0.0785 13.6730 0.7 0.0542 16.6261 0.8 0.0346 20.0576 0.9 0.0188 37.2660 1.0 0.0062 58.7784 Table 2: Numerical values of inequalities (15) for Φ(t)=et,κ= 1, and υ= 2. 2.1 Future Research Directions In this investigation, we provided a thorough examination and exhaustive derivation of integral inequalities for differentiable convex functions. We initially identified a critical integral identity that involves tempered fractional integrals. This identity is subsequently employed to deduce Weddle’s type inequalities for differentiable convex functions. Numerical examples and graphical illustrations are included to validate the significance and validity of these newly established inequalities within the tempered fractional integral perspective. Additionally, readers can 2nd Kocaeli Science Congress, November 19-21, 2025 M36-19 KOSC-2025 Proceedings investigate different fractional integrals and delve deeper into the implications of these findings across various mathematical fields. References [1] RL Burden, AM Burden, RL Burden, JD Faires, and AM Burden. Numerical analysis/richard l. Burden, J. Douglas Faires, Annette M. Burden, Cengage Learning, Boston, MA,, 2016. [2] RG Buschman. Decomposition of an integral operator by use of mikusiński calculus. SIAM Journal on Mathematical Analysis, 3(1):83–85, 1972. 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