Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new âŚ
Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.
In this article, we develop multiplicative fractional versions of Simpsonâs and Newtonâs formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using âŚ
In this article, we develop multiplicative fractional versions of Simpsonâs and Newtonâs formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature.
This paper presents a rigorous proof of novel multiplicative integral identity and utilize it to establish new Boole's type inequalities for multiplicatively convex functions. These newly established inequalities can be âŚ
This paper presents a rigorous proof of novel multiplicative integral identity and utilize it to establish new Boole's type inequalities for multiplicatively convex functions. These newly established inequalities can be helpful in finding the bounds for Boole's formula within the framework of multiplicative calculus. Moreover, Boole's type inequalities provide best optimal approximations for polynomials of degree six. Finding an error term using the first derivative is an excellent achievement in inequality theory because the class of first-time differentiable functions is more extensive than that of bounded functions with six derivatives. Numerical examples and graphical analysis are conducted to validate the effectiveness of the newly derived findings. Furthermore, the derived results are applied to the quadrature formula and special means of real numbers, demonstrating their practical utility within the context of multiplicative calculus. This research highlights their potential impact on computational mathematics and related fields. The establishment of Boole's type inequalities for multiplicatively convex functions extends our understanding of inequalities in multiplicative calculus, opening avenues for future research and applications.
This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving RiemannâLiouville fractional integrals is âŚ
This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving RiemannâLiouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the RiemannâLiouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Booleâs formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings.
The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional âŚ
The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n-times fractional Booleâs-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Booleâs-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Booleâs formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations.
The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional âŚ
The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n-times fractional Booleâs-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Booleâs-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Booleâs formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations.
This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving RiemannâLiouville fractional integrals is âŚ
This paper develops integral inequalities for first-order differentiable convex functions within the framework of fractional calculus, extending Boole-type inequalities to this domain. An integral equality involving RiemannâLiouville fractional integrals is established, forming the foundation for deriving novel fractional Boole-type inequalities tailored to differentiable convex functions. The proposed framework encompasses a wide range of functional classes, including Lipschitzian functions, bounded functions, convex functions, and functions of bounded variation, thereby broadening the applicability of these inequalities to diverse mathematical settings. The research emphasizes the importance of the RiemannâLiouville fractional operator in solving problems related to non-integer-order differentiation, highlighting its pivotal role in enhancing classical inequalities. These newly established inequalities offer sharper error bounds for various numerical quadrature formulas in classical calculus, marking a significant advancement in computational mathematics. Numerical examples, computational analysis, applications to quadrature formulas and graphical illustrations substantiate the efficacy of the proposed inequalities in improving the accuracy of integral approximations, particularly within the context of fractional calculus. Future directions for this research include extending the framework to incorporate q-calculus, symmetrized q-calculus, alternative fractional operators, multiplicative calculus, and multidimensional spaces. These extensions would enable a comprehensive exploration of Booleâs formula and its associated error bounds, providing deeper insights into its performance across a broader range of mathematical and computational settings.
This paper presents a rigorous proof of novel multiplicative integral identity and utilize it to establish new Boole's type inequalities for multiplicatively convex functions. These newly established inequalities can be âŚ
This paper presents a rigorous proof of novel multiplicative integral identity and utilize it to establish new Boole's type inequalities for multiplicatively convex functions. These newly established inequalities can be helpful in finding the bounds for Boole's formula within the framework of multiplicative calculus. Moreover, Boole's type inequalities provide best optimal approximations for polynomials of degree six. Finding an error term using the first derivative is an excellent achievement in inequality theory because the class of first-time differentiable functions is more extensive than that of bounded functions with six derivatives. Numerical examples and graphical analysis are conducted to validate the effectiveness of the newly derived findings. Furthermore, the derived results are applied to the quadrature formula and special means of real numbers, demonstrating their practical utility within the context of multiplicative calculus. This research highlights their potential impact on computational mathematics and related fields. The establishment of Boole's type inequalities for multiplicatively convex functions extends our understanding of inequalities in multiplicative calculus, opening avenues for future research and applications.
In this article, we develop multiplicative fractional versions of Simpsonâs and Newtonâs formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using âŚ
In this article, we develop multiplicative fractional versions of Simpsonâs and Newtonâs formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature.
Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new âŚ
Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.
Abstract Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. âŚ
Abstract Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions.
Fractional calculus is used to construct stress-strain relationships for viscoelastic materials. These relationships are used in the finite element analysis of viscoelastically damped structures and closed-form solutions to the equations âŚ
Fractional calculus is used to construct stress-strain relationships for viscoelastic materials. These relationships are used in the finite element analysis of viscoelastically damped structures and closed-form solutions to the equations of motion are found. The attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions.
We establish some Newton's type inequalities in the case of differentiable convex functions through the well-known RiemannâLiouville fractional integrals. Furthermore, we give an example with graph and present the validity âŚ
We establish some Newton's type inequalities in the case of differentiable convex functions through the well-known RiemannâLiouville fractional integrals. Furthermore, we give an example with graph and present the validity of the newly obtained inequalities. Finally, we give some inequalities of RiemannâLiouville fractional Newton's type for functions of bounded variation.
Abstract It is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing âŚ
Abstract It is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing an n-order derivative requires a lot of differentiation for large n, the main problem is that an error bound for an n-point Newton-Cotes quadrature is only relevant for a function that is n times differentiable, a rather stringent condition. In this paper, by defining two specific linear kernels, we resolve this problem and obtain new error bounds for all closed and open types of Newton-Cotes quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function and are very general such that they cover almost all existing results in the literature. Some illustrative examples are given in this direction.
Abstract In this study, fractional versions of Milne-type inequalities are investigated for differentiable convex functions. We present Milne-type inequalities for bounded functions, Lipschitz functions, functions of bounded variation, etc., found âŚ
Abstract In this study, fractional versions of Milne-type inequalities are investigated for differentiable convex functions. We present Milne-type inequalities for bounded functions, Lipschitz functions, functions of bounded variation, etc., found in the literature. New results are established in the area of inequalities. This article is the first to study Milne-type inequalities for fractional integrals.
A novel approach to deriving a family of quadrature formulae is presented. The first member of the new family is the corrected trapezoidal rule. The second member, a two-segment rule, âŚ
A novel approach to deriving a family of quadrature formulae is presented. The first member of the new family is the corrected trapezoidal rule. The second member, a two-segment rule, is obtained by interpolating the corrected trapezoidal rule and the Simpson one-third rule. The third member, a three-segment rule, is obtained by interpolating the corrected trapezoidal rule and the Simpson three-eights rule. The fourth member, a four-segment rule is obtained by interpolating the two-segment rule with the Boole rule. The process can be carried on to generate a whole class of integration rules by interpolating the proposed rules appropriately with the Newton-Cotes rules to cancel out an additional term in the Euler-MacLaurin error formula. The resulting rules integrate correctly polynomials of degrees less or equal to n+3 if n is even and n+2 if n is odd, where n is the number of segments of the single application rules. The proposed rules have excellent round-off properties, close to those of the trapezoidal rule. Members of the new family obtain with two additional functional evaluations the same order of errors as those obtained by doubling the number of segments in applying the Romberg integration to Newton-Cotes rules. Members of the proposed family are shown to be viable alternatives to Gaussian quadrature.
Fractional calculus is used to model the viscoelastic behavior of a damping layer in a simply supported beam. The beam is analyzed by using both a continuum formulation and a âŚ
Fractional calculus is used to model the viscoelastic behavior of a damping layer in a simply supported beam. The beam is analyzed by using both a continuum formulation and a finite element formulation to predict the transient response to a step loading. The construction of the finite element equations of motion and the resulting nontraditional orthogonality conditions for the damped mode shapes are presented. Also presented are the modified forms of matrix iteration required to calculate eigenvalues and mode shapes for the damped structure. The continuum formulation, also incorporating the fractional calculus model, is used to verify the finite element approach. The location of the poles (damping and frequency) are found to be in satisfactory agreement, as are the modal amplitudes for the first several modes.

 
 
 An estimation of remamder for Simpson's quadrature formula for mappings of bounded variation and applications in theory of special means (logarithmic mean, identric mean, etc ...) are âŚ

 
 
 An estimation of remamder for Simpson's quadrature formula for mappings of bounded variation and applications in theory of special means (logarithmic mean, identric mean, etc ...) are given. 
 
 
We firstly establish an identity involving local fractional integrals. Then, with the help of this equality, some new Newton-type inequalities for functions whose the local fractional derivatives in modulus and âŚ
We firstly establish an identity involving local fractional integrals. Then, with the help of this equality, some new Newton-type inequalities for functions whose the local fractional derivatives in modulus and their some powers are generalized convex are obtained. Some applications of these inequalities for Simpsonâs quadrature rules and generalized special means are also given.
The author first establishes an algebraic structure related to zero curvature representations and propose a new approach for calculating symmetry algebras of integrable systems. Then he deduces a hierarchy of âŚ
The author first establishes an algebraic structure related to zero curvature representations and propose a new approach for calculating symmetry algebras of integrable systems. Then he deduces a hierarchy of nonisospectral flows associated with coupled KdV systems from a spectral problem with the Laurent polynomial dependent form of the spectral parameter. Furthermore, the commutator relations of Lax operators corresponding to isospectral and nonisospectral flows are worked out according to this algebraic structure, and thus a symmetry algebra for coupled KdV systems is engendered from this general theory.
An examination is made of a graphical method and derived techniques to characterize the fractional complex modulus of simple and complex viscoelastic materials, by a fit of experimental data, as âŚ
An examination is made of a graphical method and derived techniques to characterize the fractional complex modulus of simple and complex viscoelastic materials, by a fit of experimental data, as well as from all of the various states of a material, that is, rubbery, transition, and glassy regions, rather than from that limited just to the sole transition domain, a situation that often occurs in materials investigation. Concrete results of some useful materials are illustrated. Nomenclature D β = fractional derivative of order β Einf = inflexion magnitude E(Ď) = magnitude of complex modulus E0 = static Youngâs modulus E â (Ď) = complex modulus E ďż˝ (Ď) = real part of complex modulus E �� (Ď) = imaginary part of complex modulus j 2 = â1 s = Laplace variable t = time z1, z2 = complex numbers (model parameters) Îą, β, a, b, c = model parameters (real numbers) ďż˝ =g amma function e(t) = strain e â (Ď) =F ourier transform of e(t) Ρmax = maximum loss factor
In the present paper the concepts of line and double integrals are modified to the multiplicative case. Two versions of the fundamental theorem of calculus for line and double integrals âŚ
In the present paper the concepts of line and double integrals are modified to the multiplicative case. Two versions of the fundamental theorem of calculus for line and double integrals are proved in the multiplicative case.
In the present paper we discuss multiplicative differentiation for complex-valued functions. Some drawbacks, arising with this concept in the real case, are explained satisfactorily. Some new difficulties, coming from the âŚ
In the present paper we discuss multiplicative differentiation for complex-valued functions. Some drawbacks, arising with this concept in the real case, are explained satisfactorily. Some new difficulties, coming from the complex nature of variables, are discussed and they are outreached. Multiplicative Cauchy-Riemann conditions are established. Properties of complex multiplicative derivatives are studied.
In this article, we establish some estimates of Simpson-like type integral inequalities for functions whose first derivatives in absolute value at certain powers are preinvex and obtain some inequality for âŚ
In this article, we establish some estimates of Simpson-like type integral inequalities for functions whose first derivatives in absolute value at certain powers are preinvex and obtain some inequality for the product of logarithmic preinvex. Mathematics Subject Classification: 26A51, 26D15
In this article, a new general identity for twice dierentiable functions via Riemann-Liouville fractional integrals is established. By making use of this equality, author has obtained new estimates on generalization âŚ
In this article, a new general identity for twice dierentiable functions via Riemann-Liouville fractional integrals is established. By making use of this equality, author has obtained new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities for functions whose second derivatives in absolute value at certain powers are, respectively, convex and quasi-convex functions via Riemann-Liouville fractional integrals.
In this paper we obtain some new integral inequalities like Hermite-Hadamard type for log convex functions and P functions.
In this paper we obtain some new integral inequalities like Hermite-Hadamard type for log convex functions and P functions.
In the paper, the authors establish some new integral inequalities for log-convex functions on co-ordinates.These newly-established inequalities are connected with integral inequalities of the Hermite-Hadamard type for log-convex functions on âŚ
In the paper, the authors establish some new integral inequalities for log-convex functions on co-ordinates.These newly-established inequalities are connected with integral inequalities of the Hermite-Hadamard type for log-convex functions on co-ordinates.
In recent years, a lot of research was devoted to Simpson's rule for numerical integration. In the paper we study a natural successor of Simpson's rule, namely the Boole's rule. âŚ
In recent years, a lot of research was devoted to Simpson's rule for numerical integration. In the paper we study a natural successor of Simpson's rule, namely the Boole's rule. It is the Newton-Cotes formula in the case where the interval of integration is divided into four subintervals of equal length. With computer software assistance, we prove novel error bounds for Boole's rule.
In this paper, we derived integral inequalities of Hermite-Hadamard type in the setting of multiplicative calculus for multiplicatively convex and convex functions. We also derived integral inequalities of Hermite-Hadamard type âŚ
In this paper, we derived integral inequalities of Hermite-Hadamard type in the setting of multiplicative calculus for multiplicatively convex and convex functions. We also derived integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively convex and convex functions in multiplicative calculus.
Some unweighted and weighted inequalities of Hermite-Hadamard type for log-convex functions defined on real intervals are given.
Some unweighted and weighted inequalities of Hermite-Hadamard type for log-convex functions defined on real intervals are given.
In this paper, we define interval-valued right-sided Riemann- Liouville fractional integrals. Later, we handle Hermite-Hadamard inequality and Hermite-Hadamard-type inequalities via interval-valued Riemann-Liouville fractional integrals.
In this paper, we define interval-valued right-sided Riemann- Liouville fractional integrals. Later, we handle Hermite-Hadamard inequality and Hermite-Hadamard-type inequalities via interval-valued Riemann-Liouville fractional integrals.
In this paper, some integral inequalities of Hermite-Hadamard type for multiplicatively s-convex functions are obtained. Also, some new inequalities involving multiplicative integrals are established for product and quotient of convex âŚ
In this paper, some integral inequalities of Hermite-Hadamard type for multiplicatively s-convex functions are obtained. Also, some new inequalities involving multiplicative integrals are established for product and quotient of convex and multiplicatively s-convex functions.
Abstract Pseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study âŚ
Abstract Pseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW ( k ) â type and AW ( k ) â type pseudo null curve in Minkowski 3-space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mi>E</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:math> [E_1^3 . We define helix and slant helix according to Bishop frame in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mi>E</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:math> [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.
Abstract Null cartan curves have been studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curves are not considered. In this paper, we âŚ
Abstract Null cartan curves have been studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curves are not considered. In this paper, we study weak AW ( k ) â type and AW ( k ) â type null cartan curve in Minkowski 3-space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mi>E</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:math> E_1^3 . We define helix according to Bishop frame in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mi>E</m:mi> <m:mn>1</m:mn> <m:mn>3</m:mn> </m:msubsup> </m:mrow> </m:math> E_1^3 . Furthermore, the necessary and sufficient conditions for the helices in Minkowski 3-space are obtained.
In this paper, we derive some error estimates of Simpson's second type quadrature formula for functions of bounded variation and Lipschitzian mappings. Also, similar error estimations for absolutely continuous functions âŚ
In this paper, we derive some error estimates of Simpson's second type quadrature formula for functions of bounded variation and Lipschitzian mappings. Also, similar error estimations for absolutely continuous functions whose first derivatives belong to L p [ Îł , δ ] with ( p < 1 ⤠â ) are established. Finally, with the help of the results given in this work, some Simpson's type inequalities involving special means are presented.
We study standard and nonlocal nonlinear SchrĂśdinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the âŚ
We study standard and nonlocal nonlinear SchrĂśdinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function $|q(t,x)|^2$ for the standard and nonlocal NLS equations.
In this paper, we firstly obtain two identities for multiplicative differentiable functions. Then by using these identities, we establish Ostrowski and Simpson type inequalities for multiplicative integrals. At the end âŚ
In this paper, we firstly obtain two identities for multiplicative differentiable functions. Then by using these identities, we establish Ostrowski and Simpson type inequalities for multiplicative integrals. At the end we give detail applications of our main results
The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameterized inequalities âŚ
The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameterized inequalities for the functions whose derivatives in absolute value are convex. Finally, we show that our main inequalities reduce to Ostrowski type inequalities, Simpson type inequalities and trapezoid type inequalities which are proved in earlier published papers.
Abstract In this study, we prove an identity for twice partially differentiable mappings involving the double generalized fractional integral and some parameters. By using this established identity, we offer some âŚ
Abstract In this study, we prove an identity for twice partially differentiable mappings involving the double generalized fractional integral and some parameters. By using this established identity, we offer some generalized inequalities for differentiable co-ordinated convex functions with a rectangle in the plane $\mathbb{R} ^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> . Furthermore, by special choice of parameters in our main results, we obtain several well-known inequalities such as the Ostrowski inequality, trapezoidal inequality, and the Simpson inequality for Riemann and RiemannâLiouville fractional integrals.
The physical principles of natural occurrences are frequently examined using nonlinear evolution equations (NLEEs). Nonlinear equations are intensively investigated in mathematical physics, ocean physics, scientific applications, and marine engineering. This âŚ
The physical principles of natural occurrences are frequently examined using nonlinear evolution equations (NLEEs). Nonlinear equations are intensively investigated in mathematical physics, ocean physics, scientific applications, and marine engineering. This paper investigates the Boiti-Leon-Manna-Pempinelli (BLMP) equation in (3+1)-dimensions, which describes fluid propagation and can be considered as a nonlinear complex physical model for incompressible fluids in plasma physics. This four-dimensional BLMP equation is certainly a dynamical nonlinear evolution equation in real-world applications. Here, we implement the generalized exponential rational function (GERF) method and the generalized Kudryashov method to obtain the exact closed-form solutions of the considered BLMP equation and construct novel solitary wave solutions, including hyperbolic and trigonometric functions, and exponential rational functions with arbitrary constant parameters. These two efficient methods are applied to extracting solitary wave solutions, dark-bright solitons, singular solitons, combo singular solitons, periodic wave solutions, singular bell-shaped solitons, kink-shaped solitons, and rational form solutions. Some three-dimensional graphics of obtained exact analytic solutions are presented by considering the suitable choice of involved free parameters. Eventually, the established results verify the capability, efficiency, and trustworthiness of the implemented methods. The techniques are effective, authentic, and straightforward mathematical tools for obtaining closed-form solutions to nonlinear partial differential equations (NLPDEs) arising in nonlinear sciences, plasma physics, and fluid dynamics.
Nonlinear science is a fundamental science frontier that includes research in the common properties of nonlinear phenomena. This article is devoted for the study of new extended hyperbolic function method âŚ
Nonlinear science is a fundamental science frontier that includes research in the common properties of nonlinear phenomena. This article is devoted for the study of new extended hyperbolic function method (EHFM) to attain the exact soliton solutions of the perturbed Boussinesq equation (PBE) and KdVâCauderyâDoddâGibbon (KdV-CDG) equation. We can claim that these solutions are new and are not previously presented in the literature. In addition, 2d and 3d graphics are drawn to exhibit the physical behavior of obtained new exact solutions.
From the past to the present, various works have been dedicated to Simpsonâs inequality for differentiable convex functions. Simpson-type inequalities for twice-differentiable functions have been the subject of some research. âŚ
From the past to the present, various works have been dedicated to Simpsonâs inequality for differentiable convex functions. Simpson-type inequalities for twice-differentiable functions have been the subject of some research. In this paper, we establish a new generalized fractional integral identity involving twice-differentiable functions, then we use this result to prove some new Simpsonâs-formula-type inequalities for twice-differentiable convex functions. Furthermore, we examine a few special cases of newly established inequalities and obtain several new and old Simpsonâs-formula-type inequalities. These types of analytic inequalities, as well as the methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.
<abstract><p>Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity âŚ
<abstract><p>Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type inequalities for functions whose second derivatives in absolute value are convex.</p></abstract>
The Simpson's inequality cannot be applied to a function that is twice differentiable but not four times differentiable or have a bounded fourth derivative in the interval under consideration. Loads âŚ
The Simpson's inequality cannot be applied to a function that is twice differentiable but not four times differentiable or have a bounded fourth derivative in the interval under consideration. Loads of articles are bound for twice differentiable convex functions but nothing, to the best of our knowledge, is known yet for twice differentiable exponentially convex and quasi-convex functions. In this paper, we aim to do justice to this query. For this, we prove several Simpson's type inequalities for exponentially convex and exponentially quasi-convex functions. Our findings refine, generalize and complement existing results in the literature. We regain previously known results by taking \(\alpha=0\). In addition, we also show the importance of our results by applying them to some special means of positive real numbers and to Simpson's quadrature rule. The obtained results can be extended for different kinds of convex functions.
Based on the RiemannâLiouville fractional integral, a new form of generalized Simpson-type inequalities in terms of the first derivative is discussed. Here, some more inequalities for convexity as well as âŚ
Based on the RiemannâLiouville fractional integral, a new form of generalized Simpson-type inequalities in terms of the first derivative is discussed. Here, some more inequalities for convexity as well as concavity are established. We expect that present outcomes are the generalization of already obtained results. Applications to beta, q-digamma, and Bessel functions are also provided.