Search Results (94)

Fractional differential equations have emerged as a prominent focus of modern scientific research due to their advantages in describing the complexity and nonlinear behavior of many physical phenomena. In particular, when considering problems with initial-boundary value conditions, the solution of nonlinear fractional differential equations becomes particularly important. This paper aims to explore the fractional soliton solutions for the three-component fractional nonlinear Schrödinger (TFNLS) equation under the zero background. According to the Lax pair and fractional recursion operator, we obtain fractional nonlinear equations with Riesz fractional derivatives, which ensure the integrability of these equations. In particular, by the completeness relation of squared eigenfunctions, we derive the explicit form of the TFNLS equation. Subsequently, in the reflectionless case, we construct the fractional N-soliton solutions via the Riemann–Hilbert (RH) method. The analysis results indicate that as the order of the Riesz fractional derivative increases, the widths of both one-soliton and two-soliton solutions gradually decrease. However, the absolute values of wave velocity, phase velocity, and group velocity of one component of the vector soliton exhibit an increasing trend, and show power-law relationships with the amplitude. Full article

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

A class of initial boundary value problems is here considered for a one-dimensional diffusion equation with a general time-fractional derivative with the Sonin kernel. One of the boundary conditions is in a general non-classical form, which includes no-nlocal cases of integral or multi-point boundary conditions. The problem is studied here by applying spectral projection operators to convert it to a system of relaxation equations in generalized eigenspaces. The uniqueness of the solution is established based on the uniqueness property of the spectral expansion. An algorithm is given for constructing the solution in the form of spectral expansion in terms of the generalized eigenfunctions. Estimates for the time-dependent components in this expansion are established and applied to prove the existence of a solution in the classical sense. The obtained results are applied to a particular case in which the specified boundary conditions lead to two sequences of eigenvalues, one of which consists of triple eigenvalues. Full article

This article investigates a mathematical model with the Caputo derivative for the transient unidirectional flow of an incompressible viscous fluid with pressure-dependent viscosity. The fluid flows in the spatial domain bounded by two parallel plates extended to infinity. The plates translate in their planes with time-dependent velocities, and the fluid adheres to the solid boundaries. The generalization of the model consists of formulating a fractional constitutive equation to introduce the memory effect into the mathematical model. In addition, the fluid’s viscosity is assumed to be pressure-dependent. More precisely, in this article, the viscosity is considered a power function of the vertical coordinate of the channel. Analytic solutions of the dimensionless initial and boundary value problems have been determined using the Laplace transform and Bessel equations. The inversion of Laplace transforms is conducted using both the methods of complex analysis and the Stehfest numerical algorithm. In addition, we discuss the explicit solution in some meaningful particular cases. Using numerical simulations and graphical representations, the results of the ordinary model $(\alpha =1)$ are compared with those of the fractional model $(0<\alpha <1)$, highlighting the influence of the memory parameter on fluid behavior. Full article

In this paper, we study direct and inverse problems for a spatial-fractional Black–Scholes equation with space-dependent volatility. For the direct problem, we provide CN-WSGD (Crank–Nicholson and the weighted and shifted Grünwald difference) scheme to solve the initial boundary value problem. The latter aims to recover the implied volatility via observable option prices. Using a linearization technique, we rigorously derive a mathematical formulation of the inverse problem in terms of a Fredholm integral equation of the first kind. Based on an integral equation, an efficient numerical reconstruction algorithm is proposed to recover the coefficient. Numerical results for both problems are provided to illustrate the validity and effectiveness of proposed methods. Full article

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit finite difference scheme incorporating matrix transfer technique is developed for solving the initial-boundary value problem numerically. Regarding the inverse problem, we prove the solution uniqueness and stability estimates based on interior measurement data. The source identification problem is reformulated as a variational problem using the Tikhonov regularization method, and an approximate solution to the inverse problem is obtained with the aid of the optimal perturbation algorithm. Extensive numerical simulations involving six test cases in both 1D and 2D configurations demonstrate the high effectiveness and satisfactory stability of the proposed methodology. Full article

We consider the melting of a one-dimensional domain ($x\ge 0$), initially at the melting temperature $u=0$, by fixing the boundary temperature to a value $u(0,t)=U_0>0$—the so called Stefan melting problem. The governing transient heat-conduction equation involves a time derivative and the spatial derivative of the temperature gradient. In the general case the order of the time derivative and the gradient can take values in the range $(0,1]$. In these problems it is known that the advance of the melt front $s\left(t\right)$ can be uniquely determined by a specified prefactor multiplying a power of time related to the order of the fractional derivatives in the governing equation. For given fractional orders the value of the prefactor is the unique solution to a transcendental equation formed in terms of special functions. Here, our main purpose is to provide efficient numerical schemes with low computational complexity to compute these prefactors. The values of the prefactors are obtained through a dimensionalization that allows the recovery of the solution for the quasi-stationary case when the Stefan number approaches zero. The mathematical analysis of this convergence is given and provides consistency to the numerical results obtained. Full article

In this study, we suggest a straightforward analytical/semi-analytical method based on the Elzaki transform (ET) method to find the solution to a number of differential fractional boundary value problems with initial conditions (ICs). The suggested approach not only resolves the issue of some equation nonlinearity but also transforms the issue into a simpler algebraic recurrence problem. In science and engineering, fractional differential equations (FDEs) can be solved with the help of this basic but effective concept. Some illustrative cases are used to demonstrate the efficacy and value of the suggested technique. Full article

This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves rearranging the fractional operator as a series of higher-order derivatives considered at a specific point. We demonstrate the effect of the number of terms included in the series expansion on the solution accuracy and error analysis. The advantage of the method is its simplicity and ease of implementation. Additionally, the method allows for a quick estimation of the fractional derivative by using a few first terms of the expansion. The elaborated algorithm is tested against a comprehensive series of illustrative examples, providing very good agreement with the exact/reference solutions. Furthermore, the application of the proposed method to fractional boundary/initial value problems is included. Full article

In this paper, we focus on the multiplicity of positive solutions for a singular tempered fractional initial-boundary value problem with a p-Laplacian operator and a changing-sign perturbation term. By introducing a truncation function and combing with the properties of the solution of isomorphic linear equations, we transform the changing-sign tempered fractional initial-boundary value problem into a positive problem, and then the existence results of multiple positive solutions are established by the fixed point theorem in a cone. It is worth noting that the changing-sign perturbation term only satisfies the weaker Carathèodory conditions, which implies that the perturbation term ℏ can be allowed to have an infinite number of singular points; moreover, the value of the changing-sign perturbation term can tend to negative infinity in some singular points. Full article

The primary aim of this manuscript is to establish unique fixed point results for a class of $\Psi$-contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of $\Psi$-contraction operators, we introduce a novel function, denoted as $\psi$, and explore its properties. Our work presents new theoretical results, supported by examples and applications, that enrich the study of G-metric spaces. These results not only generalize and unify a broad range of existing findings in the literature but also expand their use to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. In doing so, we offer a more comprehensive understanding of fixed point theory in the G-metric space framework and broaden its scope in applied mathematics. We also offer a numerical spectral approach for solving fractional initial value problems, utilizing shifted Chebyshev polynomials to construct a semi-analytic solution that inherently satisfies the given homogeneous initial conditions. Full article

In the literature so far, for Caputo fractional boundary value problems of order $2q$ when $1<2q<2,$ the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order $2q.$ Further, even the initial conditions for a Caputo fractional differential equation of order $nq$ use the corresponding integer-order initial conditions of order ‘n’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘$nq$’ will only be a $C^{nq}$ solution and not a $C^n$ solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order $kq$, $k=0,1,\dots (n-1)$ when the order of the fractional derivative involved in the differential equation is $nq$. The Caputo left derivative can be computed only when the function can be expressed as $f(x-a)$. Then the Caputo right derivative of the same function will be computed for the function $f(b-x).$ Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when $q=1.$ Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data. Full article

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics. Full article

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields. Full article