A Class of Integral Inequality and Application

2013 ◽  
Vol 785-786 ◽  
pp. 1395-1398 ◽  
Author(s):  
Wu Sheng Wang
Keyword(s):  
Differential Equations ◽  
Upper Bound ◽  
Integral Inequality ◽  
Unknown Function ◽  
Initial Conditions ◽  
Integral Inequalities ◽  
Inequality Technique ◽  
Bound Estimation

We discuss a class of generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retarded items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions.

Retarded Integral Inequality and its Application in Retarded Differential Equation

2012 ◽  
Vol 220-223 ◽  
pp. 2269-2272
Author(s):  
Wu Sheng Wang ◽  
Xiao Liang Zhou ◽  
Yuan Hua Lin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Social Economic ◽  
Initial Conditions ◽  
Integral Inequalities ◽  
Engineering Technology ◽  
Retarded Differential Equation ◽  
Type Integral ◽  
Social Economic Development ◽  
Bound Estimation

Differential equations are important tools in studying of natural science, engineering technology, and the laws of social economic development. It is necessary to seek some new inequalities in order to study of boundedness, uniqueness, stability and boundary value problem of a differential equation. Motivated by Abdeldaim integral inequalities, in this paper, we establish a class of generalized retarded nonlinear Gronwall-Bellman-Type integral inequalities and give upper bound estimation of the unknown function by analysis skills. Finally we give an example to illustrate the effectiveness of our results in estimation of solutions of some differential equations with the initial conditions.

scholarly journals Integral inequalities of Gronwall type for piecewise continuous functions

1997 ◽  
Vol 10 (1) ◽  
pp. 89-94 ◽  
Author(s):  
Drumi D. Bainov ◽  
Snezhana G. Hristova
Keyword(s):  
Differential Equations ◽  
Integral Inequality ◽  
Initial Value Problem ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Continuous Functions ◽  
Integral Inequalities ◽  
Initial Value ◽  
Volterra Type ◽  
Piecewise Continuous

In this paper we generalize the integral inequality of Gronwall and study the continuous dependence of the solution of the initial value problem for nonlinear impulsive integro-differential equations of Volterra type on the initial conditions.

scholarly journals Numerical and Analytical Study for Fourth-Order Integro-Differential Equations Using a Pseudospectral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
N. H. Sweilam ◽  
M. M. Khader ◽  
W. Y. Kota
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Upper Bound ◽  
Unknown Function ◽  
Approximate Formula ◽  
Analytical Study ◽  
Pseudospectral Method ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Collocation Points

A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement of the unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points to system of algebraic equations with shifted Chebyshev coefficients. Thus, by solving this system of equations, the shifted Chebyshev coefficients are obtained. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. Numerical results are performed in order to illustrate the usefulness and show the efficiency and the accuracy of the present work.

scholarly journals Integral inequalities and applications

1980 ◽  
Vol 21 (1) ◽  
pp. 13-20 ◽  
Author(s):  
K. Narsimha Reddy
Keyword(s):  
Differential Equations ◽  
Integral Inequality ◽  
Integral Inequalities

In this paper some nonlinear analogues of Gronwall's integral inequality are established and an application to differential equations is given.

Upper and Lower Bounds of Unknown Functions in Several Nonlinear Integral Inequalities in Engineering

2014 ◽  
Vol 1008-1009 ◽  
pp. 1517-1520
Author(s):  
Li Mian Zhao ◽  
Ji Ting Huang ◽  
Wu Sheng Wang
Keyword(s):  
Lower Bounds ◽  
Integral Inequality ◽  
Unknown Function ◽  
Integral Inequalities ◽  
Upper And Lower Bounds ◽  
Nonlinear Integral Inequality ◽  
Amplification Method ◽  
Lower Estimation ◽  
Change Of Variable ◽  
Linear Integral

In this paper, we discuss the upper and lower bounds of unknown functions in several nonlinear integral inequalities. Firstly, we give out the upper estimation of unknown function of a nonlinear integral inequality. Secondly, we give out the lower estimation of unknown function of another nonlinear integral inequality. Finally, we discuss the upper and lower bounds of a linear integral inequality by adopting novel analysis techniques, such as change of variable, amplification method, differential and integration.

scholarly journals Some Nonlinear Integral Inequalities Connected with Retarded Terms on Time Scales

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Zareen A. Khan ◽  
Hijaz Ahmad ◽  
Saima Rashid ◽  
Kadir Kaynak ◽  
Miao-Kun Wang
Keyword(s):  
Time Scales ◽  
Upper Bound ◽  
Unknown Function ◽  
Integral Inequalities ◽  
Specific Class ◽  
Discrete Form ◽  
Nonlinear Integrals ◽  
Upper Bound Estimate ◽  
New Inequalities

The objective of this research is to formulate a specific class of integral inequalities of Gronwall kind concerning retarded term and nonlinear integrals with time scales theory. Our results generate several new inequalities that reflect continuous and discrete form, as well as giving the unknown function an upper bound estimate. The effectiveness of such inequalities arises from the belief that it is widely relevant in unique circumstances where there is no valid utilization of various available inequalities. Applications are additionally represented to display the legitimacy of built-up hypotheses.

scholarly journals On nonlinear mixed fractional integro-differential equations with positive constant coefficient

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5623-5638 ◽  
Author(s):  
Shivaji Tate ◽  
V.V. Kharat ◽  
H.T. Dinde
Keyword(s):  
Differential Equations ◽  
Positive Constant ◽  
Integral Inequality ◽  
Constant Coefficient ◽  
Continuous Dependence ◽  
Mixed Type ◽  
Initial Conditions

In this paper, we study the existence and other properties of the solution of nonlinear mixed fractional integro-differential equations with constant coefficient. Also with the help of integral inequality of mixed type, we prove the continuous dependence of the solutions on the initial conditions.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

scholarly journals A new generalization of some quantum integral inequalities for quantum differentiable convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yi-Xia Li ◽  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Mujahid Abbas ◽  
Yu-Ming Chu
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
Quantum Integral ◽  
Special Cases

AbstractIn this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.

scholarly journals Some k-fractional extension of Grüss-type inequalities via generalized Hilfer–Katugampola derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Samaira Naz ◽  
Muhammad Nawaz Naeem ◽  
Yu-Ming Chu
Keyword(s):  
Integral Inequality ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Derivative Operator ◽  
Practical Applications ◽  
Mathematical Problems ◽  
Grüss Type Inequalities

AbstractIn this paper, we prove several inequalities of the Grüss type involving generalized k-fractional Hilfer–Katugampola derivative. In 1935, Grüss demonstrated a fascinating integral inequality, which gives approximation for the product of two functions. For these functions, we develop some new fractional integral inequalities. Our results with this new derivative operator are capable of evaluating several mathematical problems relevant to practical applications.

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