Nonlinear Fractional Differential Equations Research Articles

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.

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.

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.

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

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.

Numerical simulation of nonlinear fractional delay differential equations with Mittag-Leffler kernels

This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L1 space is presented for the studied models using the proposed scheme.

Modern technique to study Cauchy-type problem of fractional variable order differential equations with infinite delay via phase space

In this paper, we use a novel approach to study the existence, uniqueness, and stability of solutions (EUSS) to a Cauchy-type problem of nonlinear fractional differential equations (FrDiEq) of variable order with infinite delay (CPNFDEVOID). Contrary to the techniques taken in the literature, which were centered on the usage of the concept of generalized intervals and the idea of piecewise constant functions, our approach is straightforward and based on a novel fractional operator that is more appropriate and demonstrates the solvability and stability of the main problem under less restrictive presumptions. The results are achieved in this paper by using Fixed Point Theory (FPT). The application, which includes an example and supporting images, concludes the paper.

An Efficient Technique to Study Time Fractional Extended Fisher-Kolmogorov Equation

Reaction-diffusion partial differential equations are among the most widely used equations in applied mathematical modelling. In this study, we examine the solutions of one such equation, namely, the time-fractional extended Fisher-Kolmogorov equation. This equation is widely used in the study of population growth and wave propagation dynamics. The technique we consider is a combination of the natural transform and the Adomian decomposition method. Fractional derivatives are considered with singular and non-singular kernels. The existence and uniqueness of the solutions are presented. We analyse two different cases of the proposed problem to determine the validity and efficacy of the proposed scheme. Additionally, numerical simulation is shown, and the nature of the achieved solution is captured in terms of plots for various fractional orders. The outcome demonstrates that the method is straightforward, efficient, and dependable. The proposed method does not require any predetermined assumptions, linearization, perturbation, or discretization, and it prevents rounding errors. Therefore, the technique is ready to be implemented for a variety of nonlinear time fractional partial differential equations.

Separation method of semifixed variables together with integral bifurcation method for solving generalized time‐fractional thin‐film equations

It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs.

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.