Integro-differential equations of the convolution on a finite interval with Kernel having a logarithmic singularity

1996 ◽  
Vol 79 (4) ◽  
pp. 1161-1165
Author(s):  
I. V. Andronov
Keyword(s):  
Differential Equations ◽  
Finite Interval ◽  
Logarithmic Singularity

scholarly journals Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 477
Author(s):  
Snezhana Hristova ◽  
Kremena Stefanova ◽  
Angel Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Finite Interval ◽  
Initial Value ◽  
Monotone Iterative ◽  
Practical Usefulness ◽  
Fractional Delay ◽  
Fractional Differential ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.

Averaging of fuzzy differential equations on a finite interval

2012 ◽  
Vol 14 (4) ◽  
pp. 547-559
Author(s):  
A. V. Plotnikov ◽  
T. A. Komleva
Keyword(s):  
Differential Equations ◽  
Finite Interval

scholarly journals Initial value problems for first order impulsive integro-differential equations of Volterra type in Banach spaces

2007 ◽  
Vol 5 (1) ◽  
pp. 9-26 ◽  
Author(s):  
Jiang Zhu ◽  
Yajuan Yu ◽  
Vasile Postolica
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Initial Value Problems ◽  
Finite Interval ◽  
Estimation Method ◽  
Partial Ordering ◽  
Initial Value ◽  
Volterra Type ◽  
First Order ◽  
Noncompactness Measure

In this paper, we use a new method and combining the partial ordering method to study the existence of the solutions for the first order nonlinear impulsive integro-differential equations of Volterra type on finite interval in Banach spaces and for the first order nonlinear impulsive integro-differential equations of Volterra type on infinite interval with infinite number impulsive times in Banach spaces. By introducing an interim space and using progressive estimation method, some restrictive conditions on impulsive terms, used before, such as, prior estimation, noncompactness measure estimations are deleted.

scholarly journals INVESTIGATION OF SOME CLASSES OF SECOND ORDER PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS WITH A POWERLOGARITHMIC SINGULARITY IN THE KERNEL

2020 ◽  
pp. 40-54
Author(s):  
Sarvar K. ZARIFZODA ◽  
◽  
Raim N. ODINAEV ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Differential Operators ◽  
Logarithmic Singularity ◽  
Second Order ◽  
Fredholm Type ◽  
Power Singularity ◽  
Characteristic Equations ◽  
Integro Differential Equation ◽  
The Given

For a class of second-order partial integro-differential equations with a power singularity and logarithmic singularity in the kernel, integral representations of the solution manifold in terms of arbitrary constants are obtained in the class of functions vanishing with a certain asymptotic behavior. Although the kernel of the given equation is not a Fredholm type kernel, the solution of the studied equation in a class of vanishing functions is found in an explicit form. We represent a second-order integro-differential equation as a product of two first-order integro-differential operators. For these one-dimensional integro-differential operators, in the cases when the roots of the corresponding characteristic equations are real and different, real and equal and complex and conjugate, the inverse operators are found. It is found that the presence of power singularity and logarithmic singularity in the kernel affects the number of arbitrary constants in the general solution. This number, depending on the roots of the corresponding characteristic equations, can reach nine. Also, the cases when the given integro-differential equation has a unique solution are found. The correctness of the obtained results with the help of the detailed solutions of concrete examples are shown. The method of solving the given problem can be used for solving model and nonmodel integro-differential equations with a higher order power singularity and logarithmic singularity in the kernel.

Iterative techniques with computer realization for initial value problems for Riemann–Liouville fractional differential equations

2020 ◽  
Vol 26 (1) ◽  
pp. 21-47 ◽  
Author(s):  
Ravi Agarwal ◽  
A. Golev ◽  
S. Hristova ◽  
D. O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Interval ◽  
Lower And Upper Solutions ◽  
Successive Approximations ◽  
Initial Value ◽  
Practical Usefulness ◽  
Computer Environment ◽  
Fractional Differential ◽  
The Given

AbstractThe main aim of this paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Riemann–Liouville fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the given problem. A number of different cases depending on the type of lower and upper solutions are studied and various schemes for constructing successive approximations are provided. The suggested schemes are applied to some problems and their practical usefulness is illustrated.

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