Nonlinear Fractional Equations Research Articles

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Existence and Ulam stability of mild solutions for nonlinear fractional inegro-differential equations in a Banach space

Existence and Ulam stability of mild solutions for nonlinear fractional inegro-differential equations in a Banach space

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss–Weierstrass semi–groups

AbstractWe present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

Dynamics of Nonlinear Time Fractional Equations in Shallow Water Waves

Dynamics of Nonlinear Time Fractional Equations in Shallow Water Waves

Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, \[ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R}^n, \] where $n \geq 1$ , $0< s<1$ , $\omega >-\lambda _{1,s}$ , $2< p< {2n}/{(n-2s)^+}$ , $\lambda _{1,s}>0$ is the lowest eigenvalue of $(-\Delta )^s + |x|^2$ . The fractional Laplacian $(-\Delta )^s$ is characterized as $\mathcal {F}((-\Delta )^{s}u)(\xi )=|\xi |^{2s} \mathcal {F}(u)(\xi )$ for $\xi \in \mathbb {R}^n$ , where $\mathcal {F}$ denotes the Fourier transform. This solves an open question in [M. Stanislavova and A. G. Stefanov. J. Evol. Equ. 21 (2021), 671–697.] concerning the uniqueness of ground states.

A new approach for solving fractional Kundu-Eckhaus equation and fractional massive Thirring model using controlled Picard’s technique with [formula omitted]-Laplace transform

This paper introduces a novel approach called the Controlled Picard technique along with ρ-Laplace transform for simulating the solutions of the fractional Kundu–Eckhaus equation (FKEE) and time-fractional massive Thirring model (FMTM). Significant applications of these equations can be found in nonlinear optics, weakly dynamic dispersive waves of water, and the study of quantum fields. Our proposed approach merges the Controlled Picard technique with the ρ-Laplace transform, resulting in an elegant framework that effectively solves nonlinear fractional equations without the need for Lagrange multipliers or Adomian polynomials. The inclusion of a small parameter h enhances convergence and is particularly advantageous for nonlinear differential equations. We demonstrate the accuracy and convergence of our method by comparing it with other analytical approaches. In addition, we offer graphical depictions of the numerical results to demonstrate the effectiveness as well as reliability of our suggested methodology. Overall, our approach is highly efficient, accurate, and straightforward to implement.

Infinite system of nonlinear fractional integral equations in two variables via measure of noncompactness in Banach algebra $$C(I\times I,E)$$

Infinite system of nonlinear fractional integral equations in two variables via measure of noncompactness in Banach algebra $$C(I\times I,E)$$

A construction of novel soliton solutions to the nonlinear fractional Kairat-II equation through computational simulation

A construction of novel soliton solutions to the nonlinear fractional Kairat-II equation through computational simulation

A survey on fractal fractional nonlinear Kawahara equation theoretical and computational analysis

With the use of the Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo (ABC) fractal fractional differential operators, this study offers a theoretical and computational approach to solving the Kawahara problem by merging Laplace transform and Adomian decomposition approaches. We show the solution’s existence and uniqueness through generalized and advanced version of fixed point theorem. We present a precise and efficient method for solving nonlinear partial differential equations (PDEs), in particular the Kawahara problem. Through careful error analysis and comparison with precise solutions, the suggested method is validated, demonstrating its applicability in solving the nonlinear PDEs. Moreover, the comparative analysis is studied for the considered equation under the aforementioned operators.

Solving a Nonlinear Fractional Integral Equation by Fixed Point Approaches Using Auxiliary Functions Under Measure of Noncompactness

This manuscript is devoted to ensure the existence of a solution to nonlinear fractional integral equations with three variables under a measure of noncompactness. In order to accomplish our main goal, we develop a new fixed point theorem that generalizes Darbo’s fixed point theorem by utilizing a measure of noncompactness and a new contraction operator. A related tripled FP theorem is also obtained. Finally, we use this generalized Darbo’s fixed point theorem to solve a nonlinear fractional integral equation involving three variables, and an example to demonstrate our results is presented.

The direct RBF-based partition of unity method for solving nonlinear fractional parabolic equations

This paper aims to analyze a novel localized radial basis function method known as the ’direct RBF-based partition of unity method’ for solving nonlinear fractional parabolic equations. In the proposed method, the weight functions are not operated on by the differential operators, resulting in a decrease in computational cost and algorithmic complexity. Another advantage of the direct RBF-based partition of unity method compared to the standard RBF-based partition of unity method is that in the direct method, it is possible to use discontinuous weights, which is not applicable in the standard method. Furthermore, we conduct a convergence analysis of the method and derive an error bound for the local approximation. This error bound is determined by considering conditions on the eigenvalues of the Laplacian operator matrix. To ensure the accuracy and reliability of our proposed approach, we conducted a numerical simulation and provided two numerical examples for comparison. We evaluated the consistency between the theoretical and numerical results. Our comparison between the standard RBF-based partition of unity and our proposed framework shows that the direct method is more efficient computationally, but both versions have relatively the same accuracy.

Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

Abstract In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

Exploring plasma dynamics: Analytical and numerical insights into generalized nonlinear time fractional Harry Dym equation

This investigation aims to examine the resolution of the generalized nonlinear time fractional Harry Dym [Formula: see text] equation through a combined application of analytical and numerical methodologies. The primary objective is to scrutinize the equation’s behavior and present proficient methodologies for its resolution. The Khater II analytical technique and numerical frameworks, specifically the Cubic-B-spline, Quantic-B-spline, and Septic-B-spline schemes, are proposed for this purpose. The outcomes of this inquiry yield substantial revelations regarding the characteristics of [Formula: see text] equation. Both the analytical approach and numerical schemes demonstrate their efficacy in yielding precise solutions. These findings carry noteworthy implications across various disciplines, encompassing physics, mathematics, and engineering. The investigated model appears in plasma physics research on soliton theory and nonlinear waves. Soliton waves, which keep their shape and velocity while propagating, are found in plasma physics and other domains. In plasma environments, the Harry Dym equation describes these solitons and their behavior. Solitons are essential for understanding plasma dynamics, including nonlinear waves and structures. They help comprehend plasma dynamics, wave interactions, and other nonlinear processes. The principal deductions drawn from this study underscore the effectiveness and viability of the proposed techniques in resolving the [Formula: see text] equation. This research introduces innovative contributions in terms of insights and methodologies pertinent to analogous nonlinear fractional equations. Its scope encompasses nonlinear dynamics, fractional calculus, and numerical analysis.

Q-homotopy analysis method for time-fractional Newell–Whitehead equation and time-fractional generalized Hirota–Satsuma coupled KdV system

In this paper, two types of fractional nonlinear equations in Caputo sense, time-fractional Newell–Whitehead equation (FNWE) and time-fractional generalized Hirota–Satsuma coupled KdV system (HS-cKdVS), are investigated by means of the q-homotopy analysis method (q-HAM). The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions. Due to the presence of the auxiliary parameter h in this method, just a few terms of the series solution are required in order to obtain better approximation. For the sake of visualization, the numerical results obtained in this paper are graphically displayed with the help of Maple.

STUDY ON TEMPORAL-FUZZY FRACTIONAL P-KDV EQUATION WITH NON-SINGULAR MITTAG LEFFLER KERNEL

This work discusses the solution of temporal-fuzzy fractional non-linear p-KdV equations employing a singular kernel and a non-singular Mittag Leffler kernel. A novel q-homotopy analysis approach with a generalised transform is proposed to study the fuzzy time-fractional model with two distinct fractional operators, and the behaviour of the solution is studied in both crisp and uncertain cases. Consequently, the efficiency and accuracy of the proposed method have been obtained by comparing the obtained numerical results with the available results under the assumption of crisp case for α = 1 that validate the obtained results. Finally, the efficiency of the proposed fractional orders is checked with distinct fractional operators.

Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return.

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Analytical Solution of Generalized Bratu-Type Fractional Differential Equations Using the Homotopy Perturbation Transform Method

In this study, we present the generalized form of the higher-order nonlinear fractional Bratu-type equation. In this generalization, we deal with a generalized fractional derivative, which is quite useful from an application point of view. Furthermore, some special cases of the generalized fractional Bratu equation are recognized and examined. To solve these nonlinear differential equations of fractional order, we employ the homotopy perturbation transform method. This work presents a useful computational method for solving these equations and advances our understanding of them. We also plot some numerical outcomes to show the efficiency of the obtained results.

Various wave solutions to the nonlinear fractional Korteweg-de Vries Zakharov-Kuznetsov equation by a new approach to the two-variable expansion scheme

Solving nonlinear differential equations is crucial for the design, optimization, and characterization of engineering systems. However, investigating such equations poses significant challenges and requires the development of new mathematical and computational techniques. In this article, we extend the two-variable (G′/G,1/G) -expansion approach and establish scores of general analytical solutions, consisting of different functions and arbitrary parameters, to the time-fractional Korteweg–de Vries Zakharov-Kuznetsov (KdV-ZK) equation. The KdV-ZK equation provides a powerful model for modulating nonlinear waves in an assortment of physical systems, including plasma physics, atmospheric science, fluid dynamics, soliton theory, and optical fibers. The proposed technique allows extracting several novel solutions to the KdV-ZK equation, which is essential for gaining new insights into physical problems. In addition, the suggested extension shortens the calculation process and enhances computational efficiency. The obtained solitons have potential applications in several scientific and technological fields. The beta fractional derivative and the consistent wave variable are considered to restructure the stated fractional nonlinear equation. This study investigates the merits and demerits of the fractional-order derivative β on the dynamics of the system through the representation of two-dimensional graphs, which vary according to different values of β. The enhanced computational efficiency might assist the researchers in exploring a wider range of phenomena, facilitating further extensive investigations, and contributing to overall progress in the fields of science and engineering.

Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models

In this paper, we apply two different methods, namely, the G′G-expansion method and the G′G2-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation.

On the Van der Waals model on granular matters with truncated M-fractional derivative

In this work, exact solutions of the Van der Waals model (vdWm) are investigated with a new algebraic analytical method. The closed-form analysis of the vdW equation arising in the context of the fluidized granular matter is implemented under the effect of time-fractional M-derivative. The vdWm is a challenging problem in the modelling of molecules and materials. Noncovalent Van der Waals or dispersion forces are frequent and have an impact on the structure, dynamics, stability, and function of molecules and materials in biology, chemistry, materials science and physics. The auxiliary equation which is known as a direct analytical method is constructed for the nonlinear fractional equation. The process includes a transformation based on Weierstrass and Jacobi elliptic functions. Wave solutions of the model are analytically verified for the various cases. Then, graphical patterns are presented to show the physical explanation of the model interactions. The achieved solutions will be of high significance in the interaction of quantum-mechanical fluctuations, granular matter and other areas of vdWm applications.