Nonlinear Fractional Differential Equations Research Articles

ON THE PERIODIC SOLITON SOLUTIONS FOR FRACTIONAL SCHRÖDINGER EQUATIONS

In this research, we use a novel version of the Extended Direct Algebraic Method (EDAM) namely generalized EDAM (gEDAM) to investigate periodic soliton solutions for nonlinear systems of fractional Schrödinger equations (FSEs) with conformable fractional derivatives. The FSEs, which is the fractional abstraction of the Schrödinger equation, grasp notable relevance in quantum mechanics. The proposed gEDAM technique entails creating nonlinear ordinary differential equations via a fractional complex transformation, which are then solved to acquire soliton solutions. Several 3D and contour graphs of the soliton solutions reveal periodicity in the wave profiles that offer crucial perspectives into the behavior of the system. The work sheds light on the dynamics of FSEs by displaying numerous families of periodic soliton solutions and their intricate relationships. These results hold significance not only for comprehending the dynamics of FSEs but also for nonlinear fractional partial differential equation applications.

Boundary value problems for nonlinear fractional differential equations with $\psi$-Caputo fractional

In this present paper, we will envisaged the existence and uniqueness of solutions for the following boundary value problem for a nonlinear fractional differential equation involving with $\psi$-Caputo fractional derivative. Our results are proved under Banach contraction principle and Krasnoselkii's fixed point theorem.

Hyers–Ulam–Rassias stability of fractional delay differential equations with Caputo derivative

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature.

Solvability of nonlocal fractional vector functional boundary value problems with p‐Laplacian on a half‐line at resonance

In this paper, the solvability of a system of fully nonlinear Hilfer fractional differential equations on the half‐line is studied. Firstly, considering that the half‐line as the domain has the problem of insufficient compactness, we defined the space of weighting functions and normed the convergence at infinity. Secondly, by constructing appropriate operators in these Banach spaces and using the extension for the continuous theorem, some sufficient conditions for the solvability of the problem are obtained. Finally, an example is given to illustrate the main results.

A new identification of Lagrange multipliers to study solutions of nonlinear Caputo–Fabrizio fractional problems

This research aims to provide a new identification of Lagrange multipliers using a novel hybrid method based on the Khalouta transform method and the variational iteration method and to study the solutions of nonlinear fractional problems. This method is called Khalouta variational iteration method (KHVIM). The fractional derivative is considered in Caputo–Fabrizio sense. The uniqueness and convergence results are investigated using Banach fixed point theorem. The results are obtained in the form of successive approximations corresponding to the proposed problem. To demonstrate the accuracy and effectiveness of the proposed method, three different nonlinear fractional partial differential equations are provided. Approximate solutions of the fractional equations were obtained. These solutions quickly converged to exact solutions with lower computational cost. Furthermore, the method used in this study is more generalized and allows our results to be more extensive and cover several new and existing fractional problems in the literature.

Initial Value Problem for Hybrid Improved Conformable Fractional Differential Equations with Retardation and Anticipation

This manuscript is devoted to proving some results concerning the existence of solutions for a class of initial value problem for nonlinear implicit fractional Hybrid differential equations and improved conformable fractional derivative. The result is based on a fixed point theorem due to Dhage. Further, an example is provided for the justification of our main result.

On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems

Abstract The main goal of this article is to study the existence and uniqueness of periodic solutions for the implicit problem with nonlinear fractional differential equation involving the Caputo tempered fractional derivative. The proofs are based upon the coincidence degree theory of Mawhin. To show the efficiency of the stated result, two illustrative examples will be demonstrated.

Fractional dynamics and sensitivity analysis of measles epidemic model through vaccination

Measles is a highly contagious disease that mainly affects children worldwide. Even though a reliable and effective vaccination is available, there were 140,000 measles deaths worldwide in 2018, and most of them were children under the age five years. In this paper, we comprehensively investigate a novel fractional SVEIR (Susceptible-Vaccinated-Exposed-Infected-Recovered) model of the measles epidemic powered by nonlinear fractional differential equations to understand the epidemic’s dynamical behaviour. We use a non-singular Atangana-Baleanu fractional derivative to analyze the proposed model, taking advantage of non-locality. The existence, uniqueness, positivity and boundedness of the solutions are shown via concepts of fixed point theory, and we also perform the Ulam-Hyers stability of the considered model. The parameter sensitivity is discussed in the context of the variance with each parameter using 3-D graphics based on the basic reproduction number. Moreover, with the Atangana-Toufik numerical scheme, numerical findings are depicted for different fractional-order values. The presented approach produce results that are efficiently consistent and in excellent agreement with the theoretical results.

Efficient Modified Adomian Decomposition Method for Solving Nonlinear Fractional Differential Equations

Based on the application of the standard Adomian method, the current paper proposes two modification approaches for the classes of the fractional differential equations and the system of fractional differential equations, which are featured through initial-value problems. Certainly, the constructed iterative schemes for the two classes are shown to be reliable, considering a number of test problems for demonstration, and upon deploying other existing numerical approaches for contrasting. In fact, the proposed schemes are found to portray less error, rapidity, accuracy and consume less computational time among others.

Recent advances in employing the Laplace homotopy analysis method to nonlinear fractional models for evolution equations and heat-typed problems

This study stems from the growing need for effective mathematical tools to tackle nonlinear fractional evolution equations, which find wide applications in physics and engineering. Traditional methods often struggle to provide exact closed-form solutions for such equations, leading to the development of novel approaches. In this context, the combination of the homotopy analysis method with the Laplace transform presents a promising avenue for obtaining exact solutions to these complex equations. By exploring this hybrid approach, the study aims to address the existing challenges in solving nonlinear fractional models and provide a reliable method for obtaining closed-form solutions. In this way, this study focuses on rendering the coupling between the homotopy analysis approach and the Laplace transform in the search for exact closed-form solutions for the class of fractional evolution and heat-typed equations. In fact, this class of equations features some interesting nonlinear Caputo fractional partial differential equations that model dissimilar physical processes in physics and engineering. A generalized recurrent scheme is derived, which is applicable to all the members of the governing class, and further analyzed the application of this very scheme on some test fractional initial-value problems of much concern. Eventually, the method has been found to be advantageous, with various advantages, including convergence to available exact solutions. Thus, of the advantage of the current study is innovative combination of mathematical methods, the applicability of these methods to a special class of fractional equations, the development of a generalized scheme, the successful solution of prototype problems, and the practical advantages of the method.

Fractional view analysis of coupled Whitham- Broer-Kaup and Jaulent-Miodek equations

In this study, I systematically investigate the fractional Whitham-Broer-Kaup (WBK) system and the fractional Coupled Jaulent-Miodek (CJM) equation under the Caputo fractional calculus. The nonlinear fractional differential equation systems are investigated via the Aboodh transform iteration method and the Aboodh residual power series method, thus offering a thorough analytical investigation. The dynamics of the fractional WBK system are accomplished using the Aboodh transform iteration method, and the Aboodh residual power series method is employed to study the behavior of the fractional CJM equation. Using established solutions, we completely analyze their dynamics employing both symbolic computations and numerical simulations. Consequently, novel solutions are identified, and the behavior of these fractional systems in the Caputo operator sense are clarified. The results obtained show good agreement of convergence of the numerical and analytical solutions, highlighting the effectiveness of the schemes adopted for unraveling the complex dynamics of fractional nonlinear systems.

A simulation function approach for optimization by approximate solutions with an application to fractional differential equation

In this work, we study the existence and uniqueness of a common best proximity point for a pair of nonself functions that are not necessarily continuous using the simulation function. In the following, we state important common best proximity point theorems as results of the main theorems of this article. This achievement allows us to have an example that covers our main theorem but does not apply to the Banach contraction principle. Finally, an application of a nonlinear fractional differential equation to support the obtained conclusions.

Nonlocal boundary value problems for functional hybrid differential equations involving generalized w-Caputo fractional operator

In this manuscript , we establish the existence and uniqueness of solutions for boundary value problems of nonlinear hybrid fractional differential equations involving generalized $\omega-$Caputo fractional derivatives. The proofs are based on Krasnoselskii fixed point theorem and some basic concept of $\omega-$Caputo fractional analysis. As application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.

Analyzing Single and Multi-valued Nonlinear Caputo Two-Term Fractional Differential Equation With Integral Boundary Conditions

Analyzing Single and Multi-valued Nonlinear Caputo Two-Term Fractional Differential Equation With Integral Boundary Conditions

Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags

Abstract This paper deals with some qualitative properties of solutions to nonlinear neutral hybrid differential equations connected to ψ-Caputo fractional derivative with time-varying lags. First, we demonstrate the problem possesses a mild solution uniquely where the source function may have temporal singularities. Second, in some cases, we indicate that the problem possesses a unique mild solution under some weaker conditions than the previous one. Third, we also obtain a result on a global mild solution for the problem. Finally, the results are further enriched by studying a new type of Ulam–Hyers stability for the main equation. The main results are obtained by applying the nice inequality, first proposed and proven in this paper. Some befit examples are given to justify the applicability of the main results.

Impressive Exact Solitons to the Space-Time Fractional Mathematical Physics Model via an Effective Method

A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of trigonometric, hyperbolic trigonometric and mixed functions. We also discuss the effect of fractional order derivative. To validate our results, we utilized the Mathematica software. Additionally, we depict some of the obtained kink, periodic, singular, and kink-singular wave solitons, using two and three dimensional graphs. The obtained results are useful in the fields of fluid dynamics, nonlinear optics, ocean engineering and others. Furthermore, these employed techniques are not only straightforward, but also highly effective when used to solve non-linear fractional partial differential equations (FPDEs).

Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

Approximate controllability of fractional differential equations driven by Markovian switching and Lévy noise with infinite delay

Motivated by applications of nonlinear dynamical systems and based on some existing works, we introduce a general class of Atangana-Baleanu nonlinear fractional neutral differential equations driven by Markovian switching and Lévy noise with infinite delay. In this paper, we aim to derive new results on the existence of solutions as well as approximate controllability for the corresponding Atangana-Baleanu equations in Banach spaces. Under suitable conditions with the help of semigroup theory, fixed point theorem as well as stopping time argument, we prove the main results. As an application, we present two examples with numerical simulations to support our theory. The results obtained in this paper improve and extend some related conclusions on this topic.

Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems

In this paper, two problems involving nonlinear time fractional hyperbolic partial differential equations (PDEs) and time fractional pseudo hyperbolic PDEs with nonlocal conditions are presented. Collocation technique for shifted Chebyshev of the second kind with residual power series algorithm (CTSCSK-RPSA) is the main method for solving these problems. Moreover, error analysis theory is provided in detail. Numerical solutions provided using CTSCSK-RPSA are compared with existing techniques in literature. CTSCSK-RPSA is accurate, simple and convenient method for obtaining solutions of linear and nonlinear physical and engineering problems.

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.