Search Results (52)

This research proposes a method for reducing the dimension of the coefficient vector for Crank–Nicolson mixed finite element (CNMFE) solutions to solve the fourth-order variable coefficient parabolic equation. Initially, the CNMFE schemes and corresponding matrix schemes for the equation are established, followed by a thorough discussion of the uniqueness, stability, and error estimates for the CNMFE solutions. Next, a matrix-form reduced-dimension CNMFE (RDCNMFE) method is developed utilizing proper orthogonal decomposition (POD) technology, with an in-depth discussion of the uniqueness, stability, and error estimates of the RDCNMFE solutions. The reduced-dimension method employs identical basis functions, unlike standard CNMFE methods. It significantly reduces the number of unknowns in the computations, thereby effectively decreasing computational time, while there is no loss of accuracy. Finally, numerical experiments are performed for both fourth-order and time-fractional fourth-order parabolic equations. The proposed method demonstrates its effectiveness not only for the fourth-order parabolic equations but also for time-fractional fourth-order parabolic equations, which further validate the universal applicability of the POD-based RDCNMFE method. Under a spatial discretization grid $40\times 40$, the traditional CNMFE method requires $2\times {41}^2$ degrees of freedom at each time step, while the RDCNMFE method reduces the degrees of freedom to $2\times 6$ through POD technology. The numerical results show that the RDCNMFE method is nearly 10 times faster than the traditional method. This clearly demonstrates the significant advantage of the RDCNMFE method in saving computational resources. Full article

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using a powerful technique named the extended generalized Kudryashov approach, we extract different rich structured soliton solutions to these models, including bell-shaped, cuspon, parabolic soliton, singular soliton, and squeezed bell-shaped soliton. We also study the impact of fractional-order derivatives on these solutions, providing new insights into the dynamics of nonlinear models. The results are compared with the existing literature, revealing novel and distinct solutions that offer a deeper understanding of these fractional models. The results show that the implemented approach is useful, reliable, and compatible for examining fractional nonlinear evolution equations in applied science and engineering. Full article

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments. Full article

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front. Full article

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. Full article

Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts. Full article

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense. Full article

This article mainly studies the double index logarithmic nonlinear fractional $g\text{-}$Laplacian parabolic equations with the Marchaud fractional time derivatives ${\displaystyle {\partial }_t^{\alpha }}$. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional $g\text{-}$Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional $g\text{-}$Laplacian parabolic equations is studied. Full article

In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results. Full article

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity: $\left\{\begin{array}{cc}{u}_t+M\left({\left[u\right]}_s^2\right){(-\Delta )}^{s}u+{(-\Delta )}^s{u}_t={\left|u\right|}^{p-2}u\ln \left|u\right|,\hfill & \mathrm{in} \Omega \times (0,T),\hfill \\ u(x,0)=u_0\left(x\right),\hfill & \mathrm{in} \Omega ,\hfill \\ u(x,t)=0,\hfill & \mathrm{on} \mathsf{\partial }\Omega \times (0,T),\hfill \end{array}\right.$, where ${\left[u\right]}_s$ is the Gagliardo semi-norm of u, ${(-\Delta )}^s$ is the fractional Laplacian, $s\in (0,1)$, $2\lambda <p<2_s^{*}=2N/(N-2s)$, $\Omega \in {\mathbb{R}}^N$ is a bounded domain with $N>2s$, and $u_0$ is the initial function. To start with, we combined the potential well theory and Galerkin method to prove the existence of global solutions. Finally, we introduced the concavity method and some special inequalities to discuss the blowup and asymptotic properties of the above problem and obtained the upper and lower bounds on the blowup at the sublevel and initial level. Full article

In this research, we focus on the symmetry of an ancient solution for a fractional parabolic equation involving logarithmic Laplacian in an entire space. In the process of studying the property of a fractional parabolic equation, we obtained some maximum principles, such as the maximum principle of anti-symmetric function, narrow region principle, and so on. We will demonstrate how to apply these tools to obtain radial symmetry of an ancient solution. Full article

We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator A and the Caputo fractional derivative of order $\alpha \in (0,2)$ in time. The previously known representation of the mild solution to such a problem does not have a conventional variation-of-constants like form, with the propagator derived from the associated homogeneous problem. Instead, it relies on the existence of two propagators with different analytical properties. This fact limits theoretical and especially numerical applicability of the existing solution representation. Here, we propose an alternative representation of the mild solution to the given problem that consolidates the solution formulas for sub-parabolic, parabolic and sub-hyperbolic equations with a positive sectorial operator A and non-zero initial data. The new representation is solely based on the propagator of the homogeneous problem and, therefore, can be considered as a more natural fractional extension of the solution to the classical parabolic Cauchy problem. By exploiting a trade-off between the regularity assumptions on the initial data in terms of the fractional powers of A and the regularity assumptions on the right-hand side in time, we show that the proposed solution formula is strongly convergent for $t\ge 0$ under considerably weaker assumptions compared to the standard results from the literature. Crucially, the achieved relaxation of space regularity assumptions ensures that the new solution representation is practically feasible for any $\alpha \in (0,2)$ and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms. Full article

The study of fractional partial differential equations is often plagued with complicated models and solution processes. In this paper, we tackle how to simplify a specific parabolic model to facilitate its analysis and solution process. That is, we investigate a general time-fractional pricing equation, and propose new transformations to reduce the underlying model to a different but equivalent problem that is less challenging. Our procedure leads to a conversion of the model to a fractional 1 + 1 heat transfer equation, and more importantly, all the transformations are invertible. A significant result which emerges is that we prove such transformations yield solutions under the Riemann–Liouville and Caputo derivatives. Furthermore, Lie point symmetries are necessary to construct solutions to the model that incorporate the behaviour of the underlying financial assets. In addition, various graphical explorations exemplify our results. Full article

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed. Full article

In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions. Full article