scholarly journals Generalized Nonlinear Volterra-Fredholm Type Integral Inequality with Two Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Yusong Lu ◽  
Wu-Sheng Wang ◽  
Xiaoliang Zhou ◽  
Yong Huang
Keyword(s):  
Differential Equations ◽  
Integral Inequality ◽  
Inverse Function ◽  
Upper Bounds ◽  
Fredholm Type ◽  
Analysis Techniques ◽  
Amplification Method ◽  
Type Integral ◽  
Dialectical Relationship ◽  
Change Of Variable

We establish a class of new nonlinear retarded Volterra-Fredholm type integral inequalities, with two variables, where known functionwin integral functions in Q.-H. Ma and J. Pečarić, 2008 is changed into the functionsw1,w2. By adopting novel analysis techniques, such as change of variable, amplification method, differential and integration, inverse function, and the dialectical relationship between constants and variables, the upper bounds of the embedded unknown functions are estimated. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations.

scholarly journals A Nonlinear Weakly Singular Retarded Henry-Gronwall Type Integral Inequality and Its Application

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yuanhua Lin ◽  
Shanhe Wu ◽  
Wu-Sheng Wang
Keyword(s):  
Differential Equations ◽  
Singular Integral ◽  
Fractional Differential Equations ◽  
Integral Inequality ◽  
Analysis Techniques ◽  
Weakly Singular ◽  
Type Integral ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Equations With Delay

We establish a class of new nonlinear retarded weakly singular integral inequality. Under several practical assumptions, the inequality is solved by adopting novel analysis techniques, and explicit bounds for the unknown functions are given clearly. An application of our result to the fractional differential equations with delay is shown at the end of the paper.

Two Differential-Integral Inequalities in Engineering

2014 ◽  
Vol 945-949 ◽  
pp. 2463-2466
Author(s):  
Wu Sheng Wang ◽  
Chun Miao Huang
Keyword(s):  
Integral Inequalities ◽  
Analysis Techniques ◽  
Amplification Method ◽  
Dialectical Relationship ◽  
Change Of Variable ◽  
Explicit Bounds

In this paper, we establish two new differential-integral inequalities. Under several practical assumptions, the inequality is solved by adopting novel analysis techniques, such as: change of variable, amplification method, differential and integration, and the dialectical relationship between constants and variables, and explicit bounds for the unknown functions are given clearly. The derived results can be applied in the study of some practical problems in engineering.

scholarly journals A Generalized Nonlinear Volterra-Fredholm Type Integral Inequality and Its Application

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Limian Zhao ◽  
Shanhe Wu ◽  
Wu-Sheng Wang
Keyword(s):  
Integral Equations ◽  
Integral Inequality ◽  
Upper Bounds ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Type Integral

We establish a new nonlinear retarded Volterra-Fredholm type integral inequality. The upper bounds of the embedded unknown functions are estimated explicitly by using the theory of inequality and analytic techniques. Moreover, an application of our result to the retarded Volterra-Fredholm integral equations for estimation is given.

Estimation on Unknown Function of a Class of Generalized Nonlinear Volterra-Fredholm Type Difference Inequality with Iterative Summation

2014 ◽  
Vol 519-520 ◽  
pp. 883-886
Author(s):  
Wu Sheng Wang ◽  
Xiao Liang Zhou
Keyword(s):  
Difference Equation ◽  
Upper Bound ◽  
Unknown Function ◽  
Inverse Function ◽  
Fredholm Type ◽  
Difference Inequality ◽  
Amplification Method ◽  
Change Of Variable ◽  
The Difference

The main objective of this paper is to establish a class of new nonlinear Volterra-Fredholm type difference inequality with iterative summation. By technique of change of variable, amplification method, difference and summation and inverse function, Upper bound estimations of unknown functions in the difference inequality are given. Finally, we study the estimation of the solution of a class of Volterra-Fredholm difference equation by our derived results..

Several Estimations of a Class of Integral Inequality in Engineering

2014 ◽  
Vol 962-965 ◽  
pp. 2748-2751
Author(s):  
Wu Sheng Wang ◽  
Ji Ting Huang
Keyword(s):  
Singular Integral ◽  
Integral Inequality ◽  
Analysis Techniques ◽  
Amplification Method ◽  
Weakly Singular ◽  
Change Of Variable ◽  
Explicit Bounds

In this paper, we discuss a class of new nonlinear weakly singular integral inequality. Under different assumptions, the inequality is solved by analysis techniques, such as: change of variable, amplification method, and three explicit bounds for the unknown functions are given clearly.

Estimation on Unknown Function of a Class of Generalized Nonlinear Volterra-Fredholm Type Difference Inequality

2014 ◽  
Vol 889-890 ◽  
pp. 571-574
Author(s):  
Wu Sheng Wang
Keyword(s):  
Difference Equations ◽  
Upper Bound ◽  
Unknown Function ◽  
Inverse Function ◽  
Fredholm Type ◽  
Difference Inequality ◽  
Amplification Method ◽  
Change Of Variable

The main objective of this paper is to establish a class of new nonlinear Volterra-Fredholm type difference inequality. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation and inverse function. The derived results can be applied in the study of solutions of Volterra-Fredholm type difference equations.

A Class of Difference Inequality and an Application to Discrete-Time Control Systems

2014 ◽  
Vol 519-520 ◽  
pp. 1295-1298
Author(s):  
Cai Feng Li ◽  
Wu Sheng Wang
Keyword(s):  
Control Systems ◽  
Discrete Time ◽  
Difference Equations ◽  
Upper Bound ◽  
Inverse Function ◽  
Fredholm Type ◽  
Difference Inequality ◽  
Time Control ◽  
Amplification Method ◽  
Change Of Variable

The main objective of this paper is to establish a class of new nonlinear Volterra-Fredholm type sum-difference inequality with two variables. By technique of change of variable, amplification method, difference and summation and inverse function, upper bound estimations of unknown functions are given. The derived results can be applied in the study of solutions of Volterra-Fredholm type difference equations.

scholarly journals A Generalized Nonlinear Sum-Difference Inequality of Product Form

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
YongZhou Qin ◽  
Wu-Sheng Wang
Keyword(s):  
Difference Equations ◽  
Upper Bound ◽  
Unknown Function ◽  
Nonlinear Function ◽  
Inverse Function ◽  
Product Form ◽  
Discrete Inequality ◽  
Amplification Method ◽  
Dialectical Relationship ◽  
Change Of Variable

We establish a generalized nonlinear discrete inequality of product form, which includes both nonconstant terms outside the sums and composite functions of nonlinear function and unknown function without assumption of monotonicity. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation, inverse function, and the dialectical relationship between constants and variables. Using our result we can solve both the discrete inequality in Pachpatte (1995). Our result can be used as tools in the study of difference equations of product form.

A Class of Singular Volterra-Fredholm Type Difference Inequality in Engineering

2014 ◽  
Vol 571-572 ◽  
pp. 132-138
Author(s):  
Wu Sheng Wang ◽  
Chun Miao Huang
Keyword(s):  
Beta Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Jensen Inequality ◽  
Fredholm Type ◽  
Difference Inequality ◽  
Amplification Method ◽  
Change Of Variable ◽  
Explicit Bounds ◽  
The Mean

In this paper, we discuss a class of new weakly singular Volterra-Fredholm difference inequality, which is solved using change of variable, discrete Jensen inequality, Beta function, the mean-value theorem for integrals and amplification method, and explicit bounds for the unknown functions is given clearly. The derived results can be applied in the study of fractional difference equations in engineering.

scholarly journals Monte-Carlo for solving large linear systems of ordinary differential equations

2021 ◽  
Vol 8 (1) ◽  
pp. 37-48
Author(s):  
Sergey M. Ermakov ◽  
◽  
Maxim G. Smilovitskiy ◽  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Differential Equations ◽  
Integral Equations ◽  
Virtual Machines ◽  
Variable Coefficients ◽  
Fredholm Type ◽  
Type Integral ◽  
Constant Coefficients ◽  
Linear Differential

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

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