Numerical solution of fractional integro-differential equations with nonlocal conditions

In this paper, we present a numerical method for solving fractional integro-differential equations with nonlocal boundary conditions using Bernstein polynomials. Some theoretical considerations regarding fractional order derivatives of Bernstein polynomials are discussed. The error analysis is carried out and supported with some numerical examples. It is shown that the method is simple and accurate for the given problem

A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2, L∞error norms and I1, I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial condition

Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option

Active Optimal Control of the KdV Equation Using the Variational Iteration Method

The optimal pointwise control of the KdV equation is investigated with an objective of minimizing a given performance measure. The performance measure is specified as a quadratic functional of the final state and velocity functions along with the energy due to open- and closed-loop controls. The minimization of the performance measure over the controls is subjected to the KdV equation with periodic boundary conditions and appropriate initial condition. In contrast to standard optimal control or variational methods, a direct control parameterization is used in this study which presents a distinct approach toward the solution of optimal control problems. The method is based on finite terms of Fourier series approximation of each time control variable with unknown Fourier coefficients and frequencies. He's variational iteration method for the nonlinear partial differential equations is applied to the problem and thus converting the optimal control of lumped parameter systems into a mathematical programming. A numerical simulation is provided to exemplify the proposed method

Some finite difference methods for solving linear fractional KdV equation

The time-fractional Korteweg de Vries equation can be viewed as a generalization of the classical KdV equation. The KdV equations can be applied in modeling tsunami propagation, coastal wave dynamics, and oceanic wave interactions. In this study, we construct two standard finite difference methods using finite difference methods with conformable and Caputo approximations to solve a time-fractional Korteweg-de Vries (KdV) equation. These two methods are named as FDMCA and FDMCO. FDMCA utilizes Caputo's derivative and a finite-forward difference approach for discretization, while FDMCO employs conformable discretization. To study the stability, we use the Von Neumann Stability Analysis for some fractional parameter values. We perform error analysis using L1 & L∞ norms and relative errors, and we present results through graphical representations and tables. Our obtained results demonstrate strong agreement between numerical and exact solutions when the fractional operator is close to 1.0 for both methods. Generally, this study enhances our comprehension of the capabilities and constraints of FDMCO and FDMCA when used to solve such types of partial differential equations laying some ground for further research

AN EFFICIENT METHOD FOR SOLVING A DISCRETE ORTHOGONAL APPROXIMATION TO FRACTIONAL BOUNDARY VALUE PROBLEMS

In this thesis we developed a numerical method for solving a class of nonlinear fractional boundary value problems using the fractional order Legendre Tau-path following method. Theoretical and numerical analyses are presented. The numerical results showed that this method works properly and efficiently

An efficient method for the analytical study of linear and nonlinear time-fractional partial differential equations with variable coefficients

The residual power series method is effective for obtaining approximate analytical solutions to fractional-order differential equations. This method, however, requires the derivative to compute the coefficients of terms in a series solution. Other well-known methods, such as the homotopy perturbation, the Adomian decomposition, and the variational iteration methods, need integration. We are all aware of how difficult it is to calculate the fractional derivative and integration of a function. As a result, the use of the methods mentioned above is somewhat constrained. In this research work, approximate and exact analytical solutions to time-fractional partial differential equations with variable coefficients are obtained using the Laplace residual power series method in the sense of the Gerasimov-Caputo fractional derivative. This method helped us overcome the limitations of the various methods. The Laplace residual power series method performs exceptionally well in computing the coefficients of terms in a series solution by applying the straightforward limit principle at infinity, and it is also more effective than various series solution methods due to the avoidance of Adomian and He polynomials to solve nonlinear problems. The relative, recurrence, and absolute errors of the three problems are investigated in order to evaluate the validity of our method. The results show that the proposed method can be a suitable alternative to the various series solution methods when solving time-fractional partial differential equations. © 2023 Samara State Technical University. All rights reserved

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations

Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent studies suggest that fractional differential and integral operators are well suited to model physical phenomena with intrinsic memory retention and anomalous behaviour. The global property of fractional operators presents difficulties in fnding either closed-form solutions or accurate numerical solutions to fractional differential equations. In rare cases, when analytical solutions are available, they often exist only in terms of complex integrals and special functions, or as infinite series. Similarly, obtaining an accurate numerical solution to arbitrary order differential equation is often computationally demanding. Fractional operators are non-local, and so it is practicable that when approximating fractional operators, non-local methods should be preferred. One such non-local method is the spectral method. In this thesis, we solve problems that arise in the ow of non-Newtonian fluids modelled with fractional differential operators. The recurrent theme in this thesis is the development, testing and presentation of tractable, accurate and computationally efficient numerical schemes for various classes of fractional differential equations. The numerical schemes are built around the pseudo{spectral collocation method and shifted Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral methods converge geometrically, are accurate and computationally efficient. The objective of this thesis is to show, among other results, that these features are true when the method is applied to a variety of fractional differential equations. A survey of the literature shows that many studies in which pseudo-spectral methods are used to numerically approximate the solutions of fractional differential equations often to do this by expanding the solution in terms of certain orthogonal polynomials and then simultaneously solving for the coefficients of expansion. In this study, however, the orthogonality condition of the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature are used to numerically find the coefficients of the series expansions. This approach is then applied to solve various fractional differential equations, which include, but are not limited to time{space fractional differential equations, two{sided fractional differential equations and distributed order differential equations. A theoretical framework is provided for the convergence of the numerical schemes of each of the aforementioned classes of fractional differential equations. The overall results, which include theoretical analysis and numerical simulations, demonstrate that the numerical method performs well in comparison to existing studies and is appropriate for any class of arbitrary order differential equations. The schemes are easy to implement and computationally efficient