Nonlinear Fractional Differential Equations Research Articles

A shifted Chebyshev operational matrix method for pantograph‐type nonlinear fractional differential equations

In this study, we investigate and analyze an approximation of the Chebyshev polynomials for pantograph‐type fractional‐order differential equations. First, we construct the operational matrices of pantograph and Caputo fractional derivatives using Chebyshev interpolation. Then, the obtained matrices are utilized to approximate the fractional derivative. We also provide a detailed convergence analysis in terms of the weighted square norm. Finally, we describe and discuss the results of three numerical experiments conducted to confirm the applicability and accuracy of the computational scheme.

On Nonlinear Fractional Relaxation Differential Equations

On Nonlinear Fractional Relaxation Differential Equations

A robust computational analysis of residual power series involving general transform to solve fractional differential equations

In this paper, we provide a new semi-analytical approach, General Residual Power Series Method (GRPSM), to solve fractional differential equations (FDEs). This technique is simple and effective for finding an accurate and approximate solution to linear and nonlinear FDEs. Furthermore, the graphical and numerical results are described in various fractional orders. The solution obtained by GRPSM is compared with Adomian decomposition and Homotopy analysis transform method. We have solved fractional ordered gas dynamics equations and drainage equations using GRPSM, to show the applicability and simplicity of this method.

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods.

Fixed Point Results in Neutrosophic Rectangular Extended b-Metric Spaces

In this manuscript, we establish the notion of neutrosophic rectangular extended b-metric spaces and derive some fixed point results for contraction mappings. Also, we provide several non-trivial examples. Our results are more generalized with respect to the existing ones in the literature. At the end of the paper, we provide an application to non-linear fractional differential equations to test the validity of the results.

A New Hybrid Optimal Auxiliary Function Method for Approximate Solutions of Non-Linear Fractional Partial Differential Equations

This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville’s fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method’s effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques.

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

Numerical Solution for a Fractional Differential Equation Arising In Optics

A new recursive algorithm (named as Abel-ADM) is given to obtain approximate solution for a class of nonlinear fractional differential equation arising in optics. Abel-ADM is a method that combines the generalized Abel equation with the Adomian decomposition method (ADM). The effectiveness of the method was verified through numerical examples.

Analysis of traveling wave solutions of two space-time nonlinear fractional differential equations by the first-integral method

The intent of this work to implement first-integral method to study traveling wave solutions of some space-time nonlinear fractional differential equations (FDEs) and present their graphical simulations for analyzing different wave profiles. We show how a specific properties of Gamma functions and wave transformation can be used to reduce a FDE to an ordinary one. This method works well and reveals distinct exact solutions which are classified into two different types, namely trigonometric function and hyperbolic function solutions. The results are also depicted graphically in both 3D and 2D for different values of associated parameters. The obtained results may be useful to understand ion-acoustic waves in plasma, shallow water waves in seas and the evolution of a wave packet in three dimensions with finite depth on water under weak nonlinearity by the space-time-fractional regularized long wave equation and space-time-fractional Davey–Stewartson equation, respectively.

Exploring Families of Solitary Wave Solutions for the Fractional Coupled Higgs System Using Modified Extended Direct Algebraic Method

In this paper, we suggest the modified Extended Direct Algebraic Method (mEDAM) to examine the existence and dynamics of solitary wave solutions in the context of the fractional coupled Higgs system, with Caputo’s fractional derivatives. The method begins with the formulation of nonlinear differential equations using a fractional complex transformation, followed by the derivation of solitary wave solutions. Two-dimensional, Three-dimensional and contour graphs are used to investigate the behavior of traveling wave solutions. The research reveals many families of solitary wave solutions as well as their deep interrelationships and dynamics. These discoveries add to a better understanding of the dynamics of the fractionally coupled Higgs system and have potential applications in areas that use nonlinear Fractional Partial Differential Equations (FPDEs).

Group Analysis Explicit Power Series Solutions and Conservation Laws of the Time-Fractional Generalized Foam Drainage Equation

In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

Positive Solutions for Periodic Boundary Value Problems of Fractional Differential Equations with Sign-Changing Nonlinearity and Green’s Function

In this paper, a class of nonlinear fractional differential equations with periodic boundary condition is investigated. Although the nonlinearity of the equation and the Green’s function are sign-changing, the results of the existence and nonexistence of positive solutions are obtained by using the Schaefer’s fixed-point theorem. Finally, two examples are given to illustrate the main results.

Existence results for fractional neutral functional differential equations with infinite delay and nonlocal boundary conditions

In this paper, we establish sufficient criteria for ensuring the existence of solutions and uniqueness for a class of nonlinear neutral Caputo fractional differential equations supplemented with infinite delay and nonlocal boundary conditions involving fractional derivatives. The theory of infinite delay and standard fixed point theorems are employed to obtain the existence results for the given problem. Examples will be constructed to illustrate the obtained results.

On solutions of linear and nonlinear fractional differential equations with application to fractional order RC type circuits

In this paper, we solve a class of fractional differential equations of order α∈(0,1) in Caputo–Fabrizio sense. Importantly, we highlight relations between the Caputo–Fabrizio and standard first order derivatives that are very important for the understanding of Caputo–Fabrizio theory and how it can be applied. Moreover, using the exactness of differential equations concept, implicit analytical solutions of such equations are presented which are extensions and generalizations to the results found in the literature. Application to fractional order RC type circuits and other examples are given to illustrate the validity and reliability of our results and theory.

Discovery of New Exact Wave Solutions to the M-Fractional Complex Three Coupled Maccari’s System by Sardar Sub-Equation Scheme

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs).

Conformable non-polynomial spline method: A robust and accurate numerical technique

The article introduces a novel and developed a numerical technique that solves nonlinear time fractional differential equations. It combines finite difference and non-polynomial spline methods while using conformable descriptions of fractional derivatives. The strategy has proven to be useful to analyzing intricate datasets across various domains, including finance, science, and engineering. The applicability and validity of the approach are demonstrated through numerical examples. The study assessed the approach's stability using the Fourier method and found it to be unconditionally stable for a specific parameter range. The proposed method has been analyzed for convergence, and the analysis reveals that it exhibits a convergence order of six. The paper also includes graphs comparing the present solution to an analytical one. The method is tested with some examples of the model that have a wide range of applications in fields such as biology, ecology, and physics called the Burgers-Fisher equations and compared to previous approaches to demonstrate its effectiveness and applicability. The research evaluated the approach's accuracy and efficiency using the (L2 and L∞) norm errors. Also, the effects of time and fractional derivatives have been studied, and to the best of our knowledge, it has not been previously published in any academic or industry publication.

Fractional derivatives of fuzzy complex-variable functions and application to fuzzy fractional complex differential equations

In this paper, through theory of fuzzy complex analysis, we introduce some representations of r-cuts of trivial fuzzy zero, nontrivial fuzzy zero and fuzzy complex numbers in polar form on complex plane and establish some of their basic properties. Based on this, the addition and scalar multiplication operations of fuzzy complex numbers are established and the calculus of fuzzy complex-variable functions is studied. Moreover, we introduce a notion of fuzzy Ortigueira fractional derivative of fuzzy complex-variable functions and develop a theory of fuzzy fractional calculus on complex plane including some geometric features. As applications, the existence and uniqueness theorem of solutions for nonlinear fuzzy fractional complex-variable differential equations is established and some examples are provided.

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian

The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space.

Certain Fixed-Point Results via DS-Weak Commutativity Condition in Neutrosophic Metric Spaces With Application to Non-linear Fractional Differential Equations

This study demonstrates that, for the non-linear contractive conditions in Neutrosophic metric spaces, a common fixed-point theorem may be proved without requiring the continuity of any mappings. A novel commutativity requirement for mappings weaker than the compatibility of mappings is used to demonstrate the conclusion. We provide several examples to illustrate our major idea. Also, we provide an application to the non-linear fractional differential equation to show the validity of our main result.

Exact Solutions of Systems of Nonlinear Time-Space Fractional Partial Differential Equations Using an Iterative Method

Abstract Fractional partial differential equations are useful tools to describe transportation, anomalous, and non-Brownian diffusion. In the present paper, we propose the Daftardar-Gejji and Jafari method along with its error analysis for solving systems of nonlinear time–space fractional partial differential equations (PDEs). Moreover, we solve a variety of nontrivial time–space fractional systems of PDEs. The obtained solutions either occur in exact form or in the form of a series, which converges to a closed form. The proposed method is free from linearization and discretization and does not include any tedious calculations. Moreover, it is easily employable using the computer algebra system such as Mathematica, Maple, etc.