Nonlinear Fractional Differential Equations Research Articles

Coupled Systems of Nonlinear Proportional Fractional Differential Equations of the Hilfer-Type with Multi-Point and Integro-Multi-Strip Boundary Conditions

In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, the Leray–Schauder alternative and the well-known fixed-point theorem of Krasnosel’skiĭ. Finally, the main results are illustrated by constructing numerical examples.

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique.

Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq-Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves.

Stochastic bifurcations induced by Lévy noise in a fractional trirhythmic van der Pol system

Trirhythmical nature has aroused considerable interest in describing dynamic behaviors of a fractional self-sustained system. In this paper, the fractional trirhythmic system is considered and a bifurcation analysis in a stochastic fractional trirhythmic self sustained system subjected to Lévy noise perturbation is presented. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation to show a new bifurcation parameter. Then, based on the minimum mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of the damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Using the predictor–corrector simulation method, it is shown a good agreement between the fractional and the equivalent integer order system. The detailed parameter space study reveals that the multi-rhythmic properties of the oscillation of this system can be efficiently controlled by fractional order parameter and Lévy noise parameters. Finally, the analytical solutions are validated by numerical results of the Monte Carlo simulation of the original fractional modified van der Pol oscillator. The trirhythmic region can be detected in the parameter space (α,β,γ,δ) by choosing an approximate fractional order. The stability index and the noise intensity can in any case govern the dynamics of the self sustained system, but regulating the skewness parameter cannot bring about transitions between unimodal, bimodal and trimodal in this multi-rhythmic system. More bifurcations appear by adjusting the fractional order in this system. These results may be conducive to further exploring bifurcations in the real-world applications. This study sheds new insight lights on the understanding nontrivial effects of fractional derivative on a trirhythmic self sustained system.

Orbital b-metric spaces and related fixed point results on advanced Nashine–Wardowski–Feng–Liu type contractions with applications

In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results’ premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021).

Evaluation of the performance of fractional evolution equations based on fractional operators and sensitivity assessment

In this article, the nonlinear fractional Kudryashov’s equation and the space–time fractional nonlinear Tzitzeica–Dodd–Bullough (TDB) equation are solved using the new auxiliary equation method, which yields innovative analytical solutions using β and M-Truncated fractional derivatives. The fractional wave and Painlevé transformations are implemented to transform the space and time fractional nonlinear equations into a nonlinear ordinary differential equation. The model solutions are compared by utilizing the two fractional derivatives. Hyperbolic, trigonometric, rational, exponential, and other sorts of soliton solutions are discovered, and these forms of the outcomes illustrate the superiority of the method’s uniqueness. The solutions provided in this piece are sophisticated and comprehensive, and the results of the literature review are one instance of the findings. The key advantage of this approach over remedies is that it possesses higher options with flexible constraints. By plotting 3D as well as 2D graphs of the acquired results and analyzing the effect of the fractional parameter ρ on the wave profiles of the phenomenon, it has been determined that the fractional parameter significantly affects the wave profiles. The findings are carried out in such a way as to showcase the applicability and expertise of fractional derivatives and the proposed approach to evaluate several nonlinear fractional partial differential equations. Finally, the comprehensive sensitivity analysis of the proposed models is done by first transforming it into the format of a planer dynamical system using the Galilean transformation.

Multiple positive solutions and stability results for nonlinear fractional delay differential equations involving p ‐Laplacian operator

In this paper, we study a system of multiple positive solutions and stability results for nonlinear fractional delay differential equations involving ‐Laplacian operator. We derived adequate conditions to ensure that at least three nonnegative solutions exist by applying the conditions of the Leggett–Williams fixed‐point theory and some Green function properties. Due to a small change in time‐delay, we analyzed the Hyers–Ulam stability‐type of the equation. We used Riemann–Liouville fractional differential definition, and we assumed that nonzero delay . In addition, for application purpose, comprehensive examples are given to ensure the effectiveness and feasibility of the results in this paper. Our proposed equation generalize some literature in the system.

Spectral method for solving fractal fractional differential equations

Lately introduced fractal fractional Caputo - Fabrizio operator. By substituting the single kernel at the classic derivative of fractal fractional Caputo with the ordinary kernel This modern operator was derived. We introduce some beneficial characteristics relied on the qualifier of fractal fractional Caputo - Fabrizio. Here, we extend Caputo-Fabrizio for nonlinear fractal fractional differential equations. We apply Legendre operational matrix relied on this modern operator and then, we employ it to solve the differential equations determined in the sense of fractal fractional Caputo-Fabrizio. To show the simplicity and precision of the suggested technicality Some numerical examples are given.

A new numerical scheme non-polynomial spline for solving generalized time fractional Fisher equation

In this paper, a novel numerical scheme is developed using a new construct by non-polynomial spline for solving the time fractional Generalize Fisher equation. The proposed models represent bacteria, epidemics, Brownian motion, kinetics of chemicals and fuzzy systems. The basic concept of the new approach is constructing a non-polynomial spline with different non-polynomial trigonometric and exponential functions to solve fractional differential equations. The investigated method is demonstrated theoretically to be unconditionally stable. Furthermore, the truncation error is analyzed to determine the or-der of convergence of the proposed technique. The presented method was tested in some examples and compared graphically with analytical solutions for showing the applicability and effectiveness of the developed numerical scheme. In addition, the present method is compared by norm error with the cubic B-spline method to validate the efficiency and accuracy of the presented algorithm. The outcome of the study reveals that the developed construct is suitable and reliable for solving nonlinear fractional differential equations.

A New Version of the Generalized F-Expansion Method for the Fractional Biswas-Arshed Equation and Boussinesq Equation with the Beta-Derivative

In this article, a new version of the generalized F-expansion method is proposed enabling to obtain the exact solutions of the Biswas-Arshed equation and Boussinesq equation defined by Atangana’s beta-derivative. First, the new version generalized F-expansion method is introduced, and then, the exact solutions of the nonlinear fractional differential equations expressed with Atangana’s beta-derivative are given. When the results are examined, it is seen that single, combined, and mixed Jacobi elliptic function solutions are obtained. From the point of view, it is understood that the new version generalized F-expansion method can give significant results in finding the exact solutions of equations containing beta-derivatives.

Homotopy Analysis Method using Jumarie’s Approach for Multi-Dimensional Nonlinear Schrödinger Equations

In this paper, we suggest a fractional functional for the Homotopy Analysis Method (HAM) to solve the nonlinear fractional order partial differential equations with fractional order initial conditions by using the modified Riemann–Liouville fractional derivative proposed by G. Jumarie. Graphs have been plotted for different values of α , which clearly reflect the reliability of proposed scheme.

Modified conformable double Laplace–Sumudu approach with applications

In this study, we combine two novel methods, the conformable double Laplace-Sumudu transform (CDLST) and the modified decomposition technique. We use the new approach called conformable double Laplace-Sumudu modified decomposition (CDLSMD) method, to solve some nonlinear fractional partial differential equations. We present the essential properties of the CDLST and produce new results. Furthermore, five interesting examples are discussed and analyzed to show the efficiency and applicability of the presented method. The results obtained show the strength of the proposed method in solving different types of problems.

A fourth-order fractional Adams-type implicit–explicit method for nonlinear fractional ordinary differential equations with weakly singular solutions

Based on piecewise cubic interpolation polynomials, we develop a high-order numerical method with nonuniform meshes for nonlinear fractional ordinary differential equations (FODEs) with weakly singular solutions. This method is a fractional variant of the classical Adams-type implicit–explicit method widely used for integer order ordinary differential equations. We rigorously prove that the method is unconditionally convergent under the local Lipschitz condition of the nonlinear function, and when the mesh parameter is properly selected, it can achieve optimal fourth-order convergence for weakly singular solutions. We also prove the stability of the method, and discuss the applicability of the method to multi-term nonlinear FODEs and systems of multi-order nonlinear FODEs with weakly singular solutions. Numerical results are given to confirm the theoretical convergence results.

New approximation solution for time-fractional Kudryashov-Sinelshchikov equation using novel technique

In this paper, a novel method presented in [1] is applied to solve time-fractional Kudryashov-Sinelshchikov equation (KS equation), a nonlinear fractional partial differential equation (NFPDE). This method is highly effective in obtaining approximate solutions for strongly NFPDEs. The accuracy of the method is evaluated by estimating the error between the exact and approximate solutions. By applying this method, we obtain solutions for the KS equation at different values of the fractional order derivative and at different stages of time. These solutions are presented through tables and graphs, highlighting the behavior of the KS equation under various conditions.

Existence of Monotone Positive Solutions for Caputo–Hadamard Nonlinear Fractional Differential Equation with Infinite-Point Boundary Value Conditions

Numerical solutions and approximate solutions of fractional differential equations have been studied by mathematicians recently and approximate solutions and exact solutions of fractional differential equations are obtained in many kinds of ways, such as Lie symmetry, variational method, the optimal ADM method, and so on. In this paper, we obtain the positive solutions by iterative methods for sum operators. Green’s function and the properties of Green’s function are deduced, then based on the properties of Green’s function, the existence of iterative positive solutions for a nonlinear Caputo–Hadamard infinite-point fractional differential equation are obtained by iterative methods for sum operators; an example is proved to illustrate the main result.

Hermite Wavelet Method for Nonlinear Fractional Differential Equations

Nonlinear fractional differential equations (FDEs) constitute the basis for many dynamical systems in various areas of engineering and applied science. Obtaining the numerical solutions to those nonlinear FDEs has quickly gained importance for the purposes of accurate modelling and fast prototyping among many others in recent years. In this study, we use Hermite wavelets to solve nonlinear FDEs. To this end, utilizing Hermite wavelets and block-pulse functions (BPF) for function approximation, we first derive the operational matrices for the fractional integration. The novel contribution provided by this method involves combining the orthogonal Hermite wavelets with their corresponding operational matrices of integrations to obtain sparser conversion matrices. Sparser conversion matrices require less computational load, and also converge rapidly. Using the generated approximate matrices, the original nonlinear FDE is converted into an algebraic equation in vector-matrix form. The obtained algebraic equation is then solved using the collocation points. The proposed method is used to find a number of nonlinear FDE solutions. Numerical results for several resolutions and comparisons are provided to demonstrate the value of the method. The convergence analysis is also provided for the proposed method.

A New Numerical Approach for Solving the Fractional Nonlinear Multi-pantograph Delay Differential Equations

The numerical solution of the fractional nonlinear multi-pantograph delay differential equations is investigated by a new class of polynomials. These polynomials are equipped with an unknown auxiliary parameter a, which is obtained by using the collocation and least-squares methods. In this paper, the numerical solution of the fractional nonlinear multi-pantograph delay differential equation is displayed in the truncated series form. The existence and uniqueness of the solution and the error analysis are also investigated in this article. In four examples, the numerical results of the present method have been compared with other methods. For the first time, a-polynomials are used in this article to numerically solve the fractional nonlinear multi-pantograph delay differential equations, and accurate approximations have been displayed.

Existence, stability and global attractivity results for nonlinear Riemann-Liouville fractional differential equations

Existence, attractivity, and stability of solutions of a non-linear fractional differential equation of Riemann-Liouville type are proved using the classical Schauder fixed point theorem and a fixed point result due to Dhage. The results are illustrated with examples.

Different solutions to the conformable generalized (3 + 1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation arising in shallow-water waves

This paper employed the [Formula: see text]-expansion, Riccati equation, [Formula: see text]-expansion, and modified Kudryashov methods to find new exact solution sets for the conformable generalized [Formula: see text]-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation. The accuracy of the results has been demonstrated using a variety of graphical representations. These newly obtained solutions can be applied to further research and understand the dynamics of the Camassa–Holm–Kadomtsev–Petviashvili equation, which arises in ocean and water wave theory, hydrodynamics, plasma physics, nonlinear sciences, and engineering. The presented four methods are straightforward, robust, and successful in getting analytical solutions to nonlinear fractional differential equations, as the analytical results indicate.

On System of Variable Order Nonlinear p-Laplacian Fractional Differential Equations with Biological Application

The study of variable order differential equations is important in science and engineering for a better representation and analysis of dynamical problems. In the literature, there are several fractional order operators involving variable orders. In this article, we construct a nonlinear variable order fractional differential system with a p-Laplacian operator. The presumed problem is a general class of the nonlinear equations of variable orders in the ABC sense of derivatives in combination with Caputo’s fractional derivative. We investigate the existence of solutions and the Hyers–Ulam stability of the considered equation. The presumed problem is a hybrid in nature and has a lot of applications. We have given its particular example as a waterborne disease model of variable order which is analysed for the numerical computations for different variable orders. The results obtained for the variable orders have an advantage over the constant orders in that the variable order simulations present the fluctuation of the real dynamics throughout our observations of the simulations.

A new Bihari inequality and initial value problems of first order fractional differential equations

We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order alpha in (0,1). One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an L^{p}-Carathéodory condition for some p>frac{1}{alpha } (detailed definitions are given in the paper). We prove existence on an interval [0, T] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f(t, u) grows at most linearly in u, and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature.