Fractional Solution Research Articles

A Novel Solution Approach for Time-Fractional Hyperbolic Telegraph Differential Equation with Caputo Time Differentiation

In the current analysis, a specific efficient and applicable novel solution approach, based on a fractional power series technique and Laplace transform operator, is considered to predict certain accurate approximate solutions (ASs) for a time-fractional hyperbolic telegraph equation by aid of time-fractional derivatives in a Caputo sense. The solutions are obtained in a fractional Maclurian series formula by solving the original problem in the Laplace space aided by a limit concept having fewer small iterations than the classical fractional power series technique. To confirm applicability and feasibility of the proposed approach, three appropriate initial value problems are considered. Consequently, some simulations of gained outcomes are numerically and graphically implemented to support the effect of the fractional-order parameter on the geometric behavior of the obtained solutions. In addition, graphical representations are also fulfilled to verify the convergence analysis of the fractional series solutions of the classical solution. The proposed technique is therefore proposed to be a straightforward, accurate and powerful approach for handling varied time-fractional models in various physical phenomena.

Branch-and-cut algorithms for the covering salesman problem

The Covering Salesman Problem (CSP) is a generalization of the Traveling Salesman Problem in which the tour is not required to visit all vertices, as long as all vertices are covered by the tour. The objective of CSP is to find a minimum length Hamiltonian cycle over a subset of vertices that covers an undirected graph. In this paper, valid inequalities from the generalized traveling salesman problem are applied to the CSP in addition to new valid inequalities that explore distinct aspects of the problem. A branch-and-cut framework assembles exact and heuristic separation routines for integer and fractional CSP solutions. Computational experiments show that the proposed framework outperformed methodologies from literature with respect to optimality gaps. Moreover, optimal solutions were proven for several previously unsolved instances.

Analysis of fractional order model on higher institution students’ anxiety towards mathematics with optimal control theory

Anxiety towards mathematics is the most common problem throughout nations in the world. In this study, we have mainly formulated and analyzed a Caputo fractional order mathematical model with optimal control strategies on higher institution students’ anxiety towards mathematics. The non-negativity and boundedness of the fractional order dynamical system solutions have been analysed. Both the anxiety-free and anxiety endemic equilibrium points of the Caputo fractional order model are found, and the local stability analysis of the anxiety-free and anxiety endemic equilibrium points are examined. Conditions for Caputo fractional order model backward bifurcation are analyzed whenever the anxiety effective reproduction number is less than one. We have shown the global asymptotic stability of the endemic equilibrium point. Moreover, we have carried out the optimal control strategy analysis of the fractional order model. Eventually, we have established the analytical results through numerical simulations to investigate the memory effect of the fractional order derivative approach, the behavior of the model solutions and the effects of parameters on the students anxiety towards mathematics in the community. Protection and treatment of anxiety infectious students have fundamental roles to minimize and possibly to eradicate mathematics anxiety from the higher institutions.

Integrable fractional n-component coupled nonlinear Schrödinger model and fractional n-soliton dynamics

According to the integrable nonlinear model introduced by Ablowitz et al. with the Riesz fractional derivative, we discuss the inverse scattering transform, anomalous dispersion relations and squared eigenfunctions of the integrable fractional n-component coupled nonlinear Schrödinger model underlie the Riesz fractional derivative. The explicit form of this fractional model and its fractional n-soliton solution in the case of reflectionless are obtained based on the inverse scattering transform via Riemann-Hilbert problems. In particular, we analyze the effect of fractional-order exponent on its one- and two-soliton solutions based on the integrable fractional three-component coupled nonlinear Schrödinger model. These dynamic analyses play a significant role in improving the superdispersive transport of the nonlinear wave in fractional nonlinear media.

Soliton Solutions of Fractional Stochastic Kraenkel–Manna–Merle Equations in Ferromagnetic Materials

In this study, we take into account the fractional stochastic Kraenkel–Manna–Merle system (FSKMMS). The mapping approach may be used to produce various type of stochastic fractional solutions, such as elliptic, hyperbolic, and trigonometric functions. Solutions to the Kraenkel–Manna–Merle system equation, which explains the propagation of a magnetic field in a zero-conductivity ferromagnet, may provide insight into a variety of fascinating scientific phenomena. Moreover, we construct a variety of 3D and 2D graphics in MATLAB to illustrate the influence of the stochastic term and the conformable derivative on the exact solutions of the FSKMMS.

On the generation of metric TSP instances with a large integrality gap by branch-and-cut

This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. Given the very large number of constraints, we have implemented a branch-and-cut algorithm for solving IH-OPT. Then, by sampling cost vectors over the metric polytope and by solving the corresponding SEP, we can generate random fractional vertices of the SEP polytope. If we solve the IH-OPT problem for every sampled vertex using our branch-and-cut algorithm, we can select the generated TSP instance (i.e., cost vector), yielding the longest runtime for Concorde, the state-of-the-art TSP solver. Our computational results show that our method is very effective in producing challenging instances. As a by-product, we release the Hard-TSPLIB, a library of 41 small metric TSP instances which have a large integrality gap and are challenging in terms of runtime for Concorde.

Random fractional partial differential equations and solutions for water movement in soils: Theory and applications

AbstractThis paper analyses a set of random fractional partial differential equations (rfPDEs) for water movement in soils. The rfPDEs for both rigid and swelling soils are solved for both a random flux boundary condition (BC), and random concentration BC. Solutions from a random flux BC are presented for the large‐time and small‐time situations with the large‐time solution as a very simple method for determining the flux through the surface of the soil. The equation of cumulative infiltration is presented with random parameters of the rfPDE subject to a random concentration BC. The simulations using the results of the rfPDE for the two types of BCs yielded encouraging and stable results based on two sets of field data: the first set of the data was measurements at a single site while the second set was from 26 measurements in a small catchment. The results suggest that the presented procedures are very useful methods for the interpolation, extrapolation, and prediction of hydrological variables and parameters such as water content, hydraulic conductivity or the flux through the surface of the soil. The methodologies presented in this paper are able to reveal and reproduce the realistic hydrological processes in nature which are often stochastic and random.

Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition

The unified displacement function (UDF) is presented to describe the deformation behaviours of the tunnel profile along with time under the surface slope condition. Based on the discrete Fourier method, the third-order UDF in the physical plane is expanded to the Laurent series in the complex variable plane. The complex variable method is employed to derive the elastic analytical solution of stratum displacement, when the third-order UDF is taken as the displacement boundary condition of tunnel cross-section (DBCTC). The proposed elastic solution agrees well with the results of the finite element method for the consistent model, which verifies the correctness of the proposed analytical solution. Combining the corresponding principle and fractional Generalized Kelvin viscoelastic constitutive model, the fractional viscoelastic solution under the surface slope condition is determined. The time effect of stratum displacement is presented in two aspects: time-dependent DBCTC and time-dependent material parameters. The parameter analysis is performed to investigate influences of deformation modes of the third-order UDF, slope angle, tunnel radius and fractional order on the time effect of stratum vertical and horizontal displacement.

A Comparative Analysis of Fractional-Order Fokker–Planck Equation

The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial differential equations are used to represent physical processes, which are then solved analytically or numerically to examine the dynamical behaviour of the system. The new iterative approach and the Homotopy perturbation method are used in this article to solve the fractional order Fokker–Planck equation numerically. The Caputo sense is used to characterize the fractional derivatives. The suggested approach’s accuracy and applicability are demonstrated using illustrations. The proposed method’s accuracy is expressed in terms of absolute error. The proposed methods are found to be in good agreement with the exact solution of the problems using graphs and tables. The results acquired using the given approaches are also obtained at various fractional orders of the derivative. It is observed from the graphs and tables that fractional order solutions converge to an integer solution when the fractional orders approach the integer-order of the problems. The tabular and graphical view for the given problems is obtained through Maple. The presented approaches can be applied to existing non-linear fractional partial differential equations due to their accurate, simple and straightforward implementation.

Numerical solution of distributed-order time fractional Klein–Gordon–Zakharov system

In this work, the distributed-order time fractional Klein–Gordon–Zakharov system is introduced by substituting the second-order temporal derivative with a distributed-order fractional derivative. The Caputo fractional derivative is utilized to define this kind of distributed-order fractional derivative. A high accuracy approach based on the Chebyshev cardinal polynomials is established for this system. The proposed method turns the fractional system solution into an algebraic system solution by approximating the unknown solution via these cardinal polynomials and engaging their derivative matrices (that are obtained in this paper). Some test problems are considered to investigate the capability and accuracy of this approach.

INTEGRAL TRANSFORM AND FRACTIONAL KINETIC EQUATION

With the help of the Laplace and Fourier transforms, we arrive at the fractional kinetic equation's solution in this paper. Their respective solutions are given in terms of the Fox's H-function and the Mittag-Leffler function, which are also known as the generalisations and the Saigo-Maeda operator-based solution of the generalised fractional kinetic equation. The paper's findings have applications in a variety of engineering, astronomy, and physical scientific fields.

Solitary Wave Solutions of the Fractional-Stochastic Quantum Zakharov–Kuznetsov Equation Arises in Quantum Magneto Plasma

In this paper, we consider the (3 + 1)-dimensional fractional-stochastic quantum Zakharov–Kuznetsov equation (FSQZKE) with M-truncated derivative. To find novel trigonometric, hyperbolic, elliptic, and rational fractional solutions, two techniques are used: the Jacobi elliptic function approach and the modified F-expansion method. We also expand on a few earlier findings. The extended quantum Zakharov–Kuznetsov has practical applications in dealing with quantum electronpositron–ion magnetoplasmas, warm ions, and hot isothermal electrons in the presence of uniform magnetic fields, which makes the solutions obtained useful in analyzing a number of intriguing physical phenomena. We plot our data in MATLAB and display various 3D and 2D graphical representations to explain how the stochastic term and fractional derivative influence the exact solutions of the FSEQZKE.

Laplace Residual Power Series Method for Solving Three-Dimensional Fractional Helmholtz Equations

In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First, we introduce a novel method that combines the Laplace transform tool and the residual power series approach. We specifically give the specifics of how to apply the suggested approach to solve time-fractional nonlinear equations. Second, we use the FHE to evaluate the method’s efficacy and validity. Using 2D and 3D plots of the solutions, the derived and precise solutions are compared, confirming the suggested method’s improved accuracy. The results for nonfractional approximate and accurate solutions, as well as fractional approximation solutions for various fractional orders, are indicated in the tables. The relationship between the derived solutions and the actual solutions to each problem is examined, showing that the solution converges to the actual solution as the number of terms in the series solution of the problems increases. Two examples are shown to demonstrate the effectiveness of the suggested approach in solving various categories of fractional partial differential equations. It is evident from the estimated values that the procedure is precise and simple and that it can therefore be further extended to linear and nonlinear issues.

Nondegenerate solitons in the integrable fractional coupled Hirota equation

In this paper, based on the nonlinear fractional equations proposed by Ablowitz, Been, and Carr in the sense of Riesz fractional derivative, we explore the fractional coupled Hirota equation and give its explicit form. Unlike the previous nonlinear fractional equations, this type of nonlinear fractional equation is integrable. Therefore, we obtain the fractional n-soliton solutions of the fractional coupled Hirota equation by inverse scattering transformation in the reflectionless case. In particular, we analyze the one- and two-soliton solutions and prove that the fractional two-soliton can also be regarded as a linear superposition of two fractional single solitons as |t|→∞. Moreover, we obtain the nondegenerate fractional soliton solutions and give a simple analysis for them.

The weakly non-linear waves propagation for Kelvin-Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory.

The weakly nonlinear wave propagation that occurs in the presence of magnetic fields, in which energy is concentrated in a narrow band of wave-numbers in a dispersive and dissipative fluid. The main objective of this paper is to analyze the dimensional elliptic nonlinear Schrodinger equation under the influence of three different fractional operators. The generalized fractional soliton solutions and propagation of magnetohydrodynamics fluid in sort of solition will be visualized. The Conformable, and M-truncated fractional operator applied to classical evolution Schrodinger equation. In order to get the analytical closed form solution, one of the generalized approach new extended direct algebraic method is utilized. The fractional nonlinear elliptic Schrodinger equation is developed in three different fractional sense. The similarity transformation technique converted the controlling fractional system to ordinary differential equations. The fractional analytical solutions such as, plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions are obtained. The graphical 2-D and 3-D representation of the results is shown to express the propagation of fluid with the magnetic field by assuming the appropriate values of the involved parameters. The graphical performance of the obtained solution at various settings of parametric values and fractional order reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

Analysis of fractional Navier–Stokes equations

AbstractIn this study, we apply the fractional Laplace variational iteration method (FLVIM), a computer methodology for exploring fractional Navier–Stokes equation solutions. In light of the theory of fixed points and Banach spaces, this paper also explores the uniqueness and convergence of the solution of general fractional differential equations obtained by the suggested method. In addition, the fractional Laplace variational iteration method solution's error analysis is covered. The computational technique also clearly demonstrates the validity and dependability of the suggested method for solving fractional Navier‐Stokes equations. Furthermore, the obtained solutions are a perfect fit with previously established solutions.

Non-differentiable fractional odd-soliton solutions of local fractional generalized Broer-Kaup system by extending Darboux transformation

In this paper, a local fractional generalized Broer-Kaup (gBK) system is first de?rived from the linear matrix problem equipped with local space and time fractional partial derivatives, i.e, fractional Lax pair. Based on the derived fractional Lax pair, the second kind of fractional Darboux transformation (DT) mapping the old potentials of the local fractional gBK system into new ones is then established. Finally, non-differentiable frcational odd-soliton solutions of the local fractional gBK system are obtained by using two basic solutions of the derived fractional Lax pair and the established fractional DT. This paper shows that the DT can be extended to construct non-differentiable fractional soliton solutions of some local fractional non-linear evolution equations in mathematical physics.

On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems

In this paper, we establish the new forms of Riemann-type fractional integral and derivative operators. The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases. The properties of new operators like semi-group, inverse and certain others are discussed and its weighted Laplace transform is evaluated. Fractional integro-differential free-electron laser (FEL) and kinetic equations are established. The solutions to these new equations are obtained by using the modified weighted Laplace transform. The Cauchy problem and a growth model are designed as applications along with graphical representation. Finally, the conclusion section indicates future directions to the readers.

COMPUTATIONAL SOLUTIONS OF FRACTIONAL ELECTRIC SYMMETRIC CIRCUITS BY SUMUDU TRANSFORMATION

In this research, we study the Caputo fractional and constant proportional derivative numerical approximation of electrical symmetric circuits. It has been assumed that the derivative is in the order [Formula: see text]. For the fractional electrical symmetric circuits, the RC, LC, and RLC solutions are obtained by using the Sumudu transformation. We also compare the numerical simulation of each equation to its classical equivalent. We use a highly efficient integral transform to examine the impact of the power-law kernel. In our upcoming works, we will apply this to electrical circuits that are more intricate.

Numerical investigation for the fractional model of pollution for a system of lakes using the SCM based on the Appell type Changhee polynomials

<abstract><p>This article proposed a useful simulation to investigate the Liouville-Caputo fractional order pollution model's solution behavior for a network of three lakes connected by channels. A supposedly new approximation technique using the Appell type Changhee polynomials (ACPs) was used to treat the periodic and linear input models. This work employs the spectral collocation method based on the properties of the ACPs. The given technique creates a system of algebraic equations from the studied model. We verified the efficiency of the suggested technique by computing the residual error function. We compared the results to those obtained by the fourth-order Runge-Kutta method (RK4). Our findings confirmed that the technique used provides a straightforward and efficient tool to solve such problems. The key benefit of the suggested method is that it only requires a few easy steps, doesn't produce secular terms and doesn't rely on a perturbation parameter.</p></abstract>