Fractional-order Problems Research Articles

Construction of some new traveling wave solutions to the space-time fractional modified equal width equation in modern physics

Nonlinear fractional evolution equations are important for determining various complex nonlinear problems that occur in various scientific fields, such as nonlinear optics, molecular biology, quantum mechanics, plasma physics, nonlinear dynamics, water surface waves, elastic media and others. The space-time fractional modified equal width (MEW) equation is investigated in this paper utilizing a variety of solitary wave solutions, with a particular emphasis on their implications for wave propagation characteristics in plasma and optical fibre systems. The fractional-order problem is transformed into an ordinary differential equation using a fractional wave transformation approach. In this article, the polynomial expansion approach and the sardar sub-equation method are successfully used to evaluate the exact solutions of space-time fractional MEW equation. Additionally, in order to graphically represent the physical significance of created solutions, the acquired solutions are shown on contour, 3D and 2D graphs. Based on the results, the employed methods show their efficacy in solving diverse fractional nonlinear evolution equations generated across applied and natural sciences. The findings obtained demonstrate that the two approaches are more effective and suited for resolving various nonlinear fractional differential equations.

Existence of solutions to multi-point boundary value problem of fractional order on the half-line

The purpose of this paper is to establish the existence of solutions to multi-point fractional boundary value problem on an infinite interval. Using the fixed point theory, sufficient conditions are obtained that guarantee the existence of at least one solution. At the end, an example is presented to illustrate the main results

Numerical computation of fractional Bloch equation by using Jacobi operational matrix

In this work, we present a numerical scheme based on the operational matrix of fractional Caputo-Fabrizio (CF) integration for handling fractional Bloch equation (FBE) in nuclear magnetic resonance (NMR). The understanding of Bloch equation provides us a fundamental framework for describing magnetic resonance phenomena, facilitating breakthrough in diverse fields such as medical diagnostics, quantum computing and materials characterization. The non-integer order derivative and integration are presented in the Caputo-Fabrizio sense. To construct the operational matrix, Jacobi polynomial is used as a basis. The fractional Bloch equation is transformed into a set of algebraic equations by using the operational matrix. In order to examine the fractional order problem, we obtain an approximate solution for FBE and present the numerical results in graphical and tabular forms.

Existence and uniqueness of Solution for Boundary Value Problem of Fractional Order

In this study, we investigate a class of fractional ordering and fractional derivative-based boundary value problems. and . There are four boundary value requirements in this equation. The Banach fixed point theorem (Contraction mapping theorem) and the Schauder fixed point theorem are both used to arrive at the existence and uniqueness solution. Examples based on the fractional integral method and integral operator are used to illustrate our main points.

Novel derivative operational matrix in Caputo sense with applications

The main objective of this study is to present a computationally efficient numerical method for solving fractional-order differential equations with initial conditions. The proposed method is based on the newly developed generalized derivative operational matrix and generalized integral operational matrix derived from Laguerre polynomials, which belong to the class of orthogonal polynomials. Through the utilization of these operational matrices, the fractional-order problems can be transformed into a system of Sylvester-type matrix equations. This system is easily solvable using any computational software, thereby providing a practical framework for solving such equations. The results obtained are compared against various benchmarks, including an existing exact solution, Podlubny numerical techniques, analytical and numerical solvers, and reported solutions from stochastic techniques employing hybrid approaches. This comparative analysis serves to validate the accuracy of our proposed design scheme.

Fractional Block Method for the Solution of Fractional Order Differential Equations

The construction of the fourth-order 2-point Fractional Block Backward Differentiation Formula (2FBBDF(4)) to solve the fractional order differential equations (FDEs) is presented in this paper. The method is developed using the fractional linear multistep method (FLMM) linked with the linear difference operator. This paper aims to approximate the fractional order problems via 2FBBDF(4), which is normally used to solve ordinary differential equations. The criteria for the stability of the method are analyzed in order to solve FDE problems. Consequently, the method is determined to be \textit{A}-stable for different values of α within the interval (0,1) . Then, Newton's iteration is implemented in this method to solve the problems. Multiple numerical examples of linear, nonlinear, and system FDEs are provided for the scenario where the order α lies within the range of 0 and 1 . Ultimately, the numerical results confirm that the proposed method performs at a similar level to the existing methods.

Hamilton‐Jacobi‐Bellman equation based on fractional random impulses system

AbstractIn this talk, we consider the optimal control problem for fractional order random impulses differential equations (), and there is no corresponding Hamilton‐Jacobi‐Bellman (HJB) equation proposed for fractional order differential equations in existing literature. The lack of semi group properties in fractional order makes the original dynamic programming methods unable to directly handle fractional order problems. We have dealt with this problem by combining the properties of fractional order integrals. In solving this problem, we found that the order of the original system in HJB equation is at least 1. When the order is less than 1, our approach to fractional order ideas provides a possibility to solve the problem.

Numerical investigation of systems of fractional partial differential equations by new transform iterative technique

<p>This research introduced a new method, the Aboodh Tamimi Ansari transform method ($ (AT)^2 $ method), for solving systems of linear and nonlinear fractional partial differential equations. The method combined the Aboodh transform method and the Tamimi Ansari method, allowing for the simultaneous solution of linear and nonlinear terms without restrictions. The Caputo sense was considered for fractional derivatives. The effectiveness of the proposed method was demonstrated through numerical solutions, graphical representations, and tabular data, showing strong agreement with exact solutions. The approach was deemed precise, easy to apply, and could be extended to address further challenges in fractional-order problems. Computational tasks were carried out using Mathematica 13.</p>

Study of Fractional-Order Boundary Value Problems Using Nondiscretization Numerical Method

This paper is devoted to present a numerical scheme based on operational matrices to compute approximate solutions to fractional-order boundary value problems. For the mentioned operational matrices, we utilize fractional-order Bernoulli polynomials. Since fractional-order problems are usually difficult to treat for their corresponding analytical or exact solutions, therefore, we need sophisticated methods to find their best numerical solutions. The presented numerical scheme has the ability to reduce the proposed problem to the corresponding algebraic equations. The obtained algebraic equations are then solved by using the computational software MATLAB for the corresponding numerical results. The used method has the ability to save much more time and also is reliable to secure the proper amount of memory. Several examples are solved by using the considered method. Also, the solutions are compared with their exact solution graphically. In addition, the absolute errors for different scale values are presented graphically. In addition, we compare our results with the results of the shifted Legendre polynomials spectral method.

A Semi-Analytical Method for Solving Nonlinear Fractional-Order Swift-Hohenberg Equations

In this study, we find approximate series solutions to fractional-order Swift-Hohenberg equations by using the hybrid method, i.e., accelerated homotopy perturbation transformation method (AHPTM). The accelerated homotopy perturbation method was merged with the Laplace transform to create the proposed method. We also compare the results of our proposed method with the exact solution and demonstrate that it is the useful tool for tackling nonlinear problems of fractional order. Results are presented through graphs using Mathematica software.

Spectral analysis of variable-order multi-terms fractional differential equations

Abstract In this work, a numerical scheme based on shifted Jacobi polynomials (SJPs) is deduced for variable-order fractional differential equations (FDEs). We find numerical solution of consider problem of fractional order. The proposed numerical scheme is based on operational matrices of variable-order differentiation and integration. To create the mentioned operational matrices for variable-order integration and differentiation, SJPs are used. Using the aforementioned operational matrices, we change the problem under consideration into matrix equation. The resultant matrix equation is solved by using Matlab, which executes the Gauss elimination method to provide the necessary numerical solution. The technique is effective and produced reliable outcomes. To determine the effectiveness of the suggested method, the results are compared to the outcomes of some other numerical procedure. Additional examples are included in this article to further clarify the process. For various scale levels and fractional-order values, absolute errors are also recorded.

Semi-Analytical Solutions for Some Types of Nonlinear Fractional-Order Differential Equations Based on Third-Kind Chebyshev Polynomials

Approximate solutions for a family of nonlinear fractional-order differential equations are introduced in this work. The fractional-order operator of the derivative are provided in the Caputo sense. The third-kind Chebyshev polynomials are discussed briefly, then operational matrices of fractional and integer-order derivatives for third-kind Chebyshev polynomials are constructed. These obtained matrices are a critical component of the proposed strategy. The created matrices are used in the context of approximation theory to solve the stated problem. The fundamental advantage of this method is that it converts the nonlinear fractional-order problem into a system of algebraic equations that can be numerically solved. The error bound for the suggested technique is computed, and numerical experiments are presented to verify and support the accuracy and efficiency of the proposed method for solving the class of nonlinear multi-term fractional-order differential equations.

New analytical technique to solve fractional-order Sharma–Tasso–Olver differential equation using Caputo and Atangana–Baleanu derivative operators

Abstract The present work introduces a novel approach, the Adomian Decomposition Formable Transform Method (ADFTM), and its application to solve the fractional order Sharma-Tasso-Olver problem. The method’s distinctive outcomes are highlighted through a comparative analysis with established non-local Caputo fractional derivatives and the non-singular Atangana–Baleanu (ABC) fractional derivatives. To provide a comprehensive understanding, the proposed ADFTM’s approximate solution is compared with the homotopy perturbation method (HPM) and residual power series method (RPSM). Further, numerical and graphical results demonstrate the reliability and accuracy of the ADFTM approach. The novel outcomes presented in this work emphasize its capability to address complex engineering problems effectively. By demonstrating its efficacy in solving the fractional order problems, the new ADFTM proves to be a valuable tool in solving scientific problems.

Variational Iteration Approach for Solving Fractional Integro-Differential Equations with Conformable Differintegrals

Fractional integro-differential equations involving conformable differintegrals have promising applications but lack solution methods. This work develops a variational iteration approach for such equations. Conformable fractional derivatives and integrals that generalize integer-order equivalents are defined. Existence and uniqueness results are established for a class of nonlinear fractional integro-differential equations. These exist under less restrictive smoothness assumptions than integer-order cases. The variational iteration method (VIM) is then applied, providing an analytical approximate technique for integro-differential problems of fractional order. Convergence analysis demonstrates the efficiency and accuracy of the VIM solutions. Several test cases validate the VIM, matching analytical solutions available for simpler fractional differential sub-cases. The proposed technique advances available methods for this emerging class of fractional integro-differential equations. Significantly, it enables application of such models by allowing accurate solution of associated mathematical system representations. Extensions to include more singular equation classes, comparisons with other methods, and real-world applications are suggested as future work.
 The abstract summarizes the key points: significance of problem, fractional models used, existence/uniqueness proofs, VIM approach and analysis, test case validations, implications for applications

Analysis of age wise fractional order problems for the Covid-19 under non-singular kernel of Mittag-Leffler law

The developed article considers SIR problems for the recent COVID-19 pandemic, in which each component is divided into two subgroups: young and adults. These subgroups are distributed among two classes in each compartment, and the effect of COVID-19 is observed in each class. The fractional problem is investigated using the non-singular operator of Atangana Baleanu in the Caputo sense (ABC). The existence and uniqueness of the solution are calculated using the fundamental theorems of fixed point theory. The stability development is also determined using the Ulam-Hyers stability technique. The approximate solution is evaluated using the fractional Adams-Bashforth technique, providing a wide range of choices for selecting fractional order parameters. The simulation is plotted against available data to verify the obtained scheme. Different fractional-order approximations are compared to integer-order curves of various orders. Therefore, this analysis represents the recent COVID-19 pandemic, differentiated by age at different fractional orders. The analysis reveals the impact of COVID-19 on young and adult populations. Adults, who typically have weaker immune systems, are more susceptible to infection compared to young people. Similarly, recovery from infection is higher among young infected individuals compared to infected cases in adults.

Solution of Fractional Heat-Like and Fractional Wave-Like Equation by Using Modern Strategy

Abstract This paper introduces a novel form of the Adomian decomposition (ADM) method for solving fractional-order heat-like and wave-like equations with starting and boundary value problems. The derivations are provided in the sense of Caputo. In order to help understanding, the generalised formulation of the current approach is provided. Several numerical examples of fractional-order diffusion-wave equations (FDWEs) are solved using the suggested method in this context. In addition to examining the applicability of the suggested method to the solving of fractional-order heat-like and wave-like equations, a graphical depiction of the solutions to three instructive cases was constructed. Solution graphs were arrived at for integer and fractional-order problems. The derived and exact solutions to integer-order problems were found to be in excellent agreement. The subject of the present research endeavour is the convergence of fractional-order solutions. This strategy is considered to be the most successful way of addressing fractional-order initial-boundary value issues in science and engineering. This strategy is presented here.

Successive approximations and interval halving for fractional BVPs with integral boundary conditions

We study a system of non-linear fractional differential equations, subject to integral boundary conditions. We use a parametrization technique and a dichotomy-type approach to reduce the original problem to two “model-type” fractional boundary value problems with linear two-point boundary conditions. A numerical-analytic technique is applied to analytically construct approximate solutions to the “model-type” problems. The behaviour of these approximate solutions is governed by a set of parameters, whose values are obtained by numerically solving a system of algebraic equations. The obtained results are confirmed by an example of the fractional order problem that in the case of the second order differential equation models the Antarctic Circumpolar Current.

A Comparative Study of the Fractional Partial Differential Equations via Novel Transform

In comparison to fractional-order differential equations, integer-order differential equations generally fail to properly explain a variety of phenomena in numerous branches of science and engineering. This article implements efficient analytical techniques within the Caputo operator to investigate the solutions of some fractional partial differential equations. The Adomian decomposition method, homotopy perturbation method, and Elzaki transformation are used to calculate the results for the specified issues. In the current procedures, we first used the Elzaki transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. For each targeted problem, the generalized schemes of the suggested methods are derived under the influence of each fractional derivative operator. The current approaches give a series-form solution with easily computable components and a higher rate of convergence to the precise solution of the targeted problems. It is observed that the derived solutions have a strong connection to the actual solutions of each problem as the number of terms in the series solution of the problems increases. Graphs in two and three dimensions are used to plot the solution of the proposed fractional models. The methods used currently are simple and efficient for dealing with fractional-order problems. The primary benefit of the suggested methods is less computational time. The results of the current study will be regarded as a helpful tool for dealing with the solution of fractional partial differential equations.

Fractional-order problem coupled with a second-order Moreau’s sweeping process

Fractional-order problem coupled with a second-order Moreau’s sweeping process

Intelligent neuro-computing to analyze the awareness programs of fractional epidemic system outbreaks

A new model of fractional for the influence of consciousness initiatives on epidemic incidences (ICIEI) is described in this research. The fractional Caputo operator (FCO) is used to expand this system to the typical awareness program model (ICIEI-FCO). Analytically and quantitatively, the properties of the new system are investigated. This article presents an artificial neural network-based fractional model (ANN-FM) for estimating the efficacy of epidemic outbreak awareness initiatives. The reliability, resolution, durability, and robustness of the proposed model are examined using the suggested ANN-BLM approach for five distinct situations. The population of the model is created by utilizing the power of the explicit Runge-Kutta numerical approach, and it is represented by a system of nonlinear ordinary differential equations. The Grunwald–Letnikov (GL) technique is used to numerically evaluate the modeled differential system of the physical issue for multiple scenarios to anticipate numerical data, and these results are utilized as a reference dataset of the networks. The data needed to answer the fractional model's questions on how awareness campaigns affect epidemic outbreaks is broken down as follows: training takes up 80% of the time, testing 10%, and authorization 10%. There are two aspects to the strategy: First, the fundamental ANN-BLM operator performances are displayed. In the meantime, the ANN-BLM execution approach is used to address the fractional-order problem. The GL-mathematical system was used to compare the numerical findings. The TSs, regression, correlation, EHs, and MSE are used to demonstrate the dependability and competency of ANN-BLM as well as their numerical performances in the presented numerical findings, which were developed using ANN-BLM to lower the MSE. The Levenberg-Marquart training (LMT) algorithm is used to optimize network results in terms of mean-square errors, training states graphs, errors of the histogram, recession analysis, auto-correlation and time series responses, which demonstrates the system's accurate and proficient trend acknowledgment. The mean-square error fitness analysis in the ranges of 10-6 to 10-11 validates the authenticity and effectiveness of the designed solver. The suggested AI-based study is expected to pave the way for new, creative approaches to fractional order modeling and analysis of naturally unpredictable dynamic systems.