Related Experiment Video
Updated: May 12, 2026
06:37
Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package
Published on: September 17, 2021
Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed
Igor Podlubny1, Tomas Skovranek, Blas M Vinagre Jara
1BERG Faculty, Technical University of Kosice, B. Nemcovej 3, 04200 Kosice, Slovakia. igor.podlubny@tuke.sk
Summary
This study introduces a novel variable-step-length method for fractional calculus, enhancing numerical solutions for differential equations with fractional derivatives. The approach efficiently handles non-uniformly sampled signals, advancing fractional calculus applications.
Area of Science:
- Numerical Analysis
- Fractional Calculus
- Applied Mathematics
Background:
- Podlubny's matrix approach provides a method for discretizing fractional integrals and derivatives.
- Existing methods often struggle with non-equidistant data and variable step lengths.
Purpose of the Study:
- To develop a variable-step-length numerical method for fractional calculus.
- To extend the matrix approach to handle non-equidistant grids and distributed-order derivatives.
- To enable the numerical solution of fractional differential equations with adaptive step sizes.
Main Methods:
- Further development of Podlubny's matrix approach for fractional calculus.
- Introduction of numerical integration and differentiation on non-equidistant grids.
- Presentation of a novel 'method of large steps' for variable-step-length computations.
Main Results:
- Demonstrated numerical solutions for differential equations with constant and distributed-order fractional derivatives.
- Successful application of the matrix approach on non-equidistant grids.
- First presentation of a variable-step-length method ('method of large steps') for fractional calculus.
Conclusions:
- The developed method allows for fractional-order and distributed-order differentiation and integration of non-uniformly sampled signals.
- This work paves the way for variable- and adaptive-step-length techniques in solving fractional and distributed-order differential equations.
- The 'method of large steps' offers a new computational strategy for complex fractional calculus problems.
Related Concept Videos
01:28Partial Fractions
A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
01:20Midpoint Rule
Approximating areas under curved boundaries is a common problem in applied mathematics, particularly when an exact calculation is difficult or impractical. One effective numerical method for this purpose is the Midpoint Rule, which provides an estimate of the area under a curve by using rectangular approximations over a specified interval.Description of the Midpoint RuleThe Midpoint Rule begins by dividing the given interval into a number of equal subintervals. For each subinterval, the...
01:24Fast Decoupled and DC Powerflow
The fast decoupled power flow method addresses contingencies in power system operations, such as generator outages or transmission line failures. This method provides quick power flow solutions, essential for real-time system adjustments. Fast decoupled power flow algorithms simplify the Jacobian matrix by neglecting certain elements, leading to two sets of decoupled equations:
01:24Difference Equation Solution using z-Transform
The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
01:28Improper Integrals: Discontinuous Integrands
Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
01:24Approximate Integration
In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...