Space Derivatives Research Articles

Study of two-dimensional nonlinear coupled time-space fractional order reaction advection diffusion equations using shifted Legendre-Gauss-Lobatto collocation method

In this article, the nonlinear coupled two-dimensional space-time fractional order reaction-advection–diffusion equations (2D-STFRADEs) with initial and boundary conditions is solved by using Shifted Legendre-Gauss-Lobatto Collocation method (SLGLCM) with fractional derivative defined in Caputo sense. The SLGLC scheme is used to discretize the coupled nonlinear 2D-STFRADEs into the shifted Legendre polynomial roots to convert it to a system of algebraic equations. The efficiency and efficacy of the scheme are confirmed through error analysis while applying the scheme on two existing problems having exact solutions. The impact of advection and reaction terms on the solution profiles for various space and time fractional order derivatives are shown graphically for different particular cases. A drive has been made to study the convergence of the proposed scheme, which has been applied on the proposed mathematical model.

A split-step finite element method for the space-fractional Schrödinger equation in two dimensions

In this article, we propose a split-step finite element method (FEM) for the two-dimensional nonlinear Schrödinger equation (NLS) with Riesz fractional derivatives in space. The space-fractional NLS is first spatially discretized by finite element scheme and the semi-discrete variational scheme is obtained. We prove that it maintains the mass and energy conservation laws. Then, we establish a fully discrete split-step finite element scheme for the considered problem, which avoids the iteration at each time layer, thereby significantly reducing computational cost. The discrete mass conservation property and error estimate of this split-step finite element scheme is derived. Finally, illustrative tests and the numerical simulation of dynamic of wave solutions are included to confirm its effectiveness and capability.

Tempered fractional differential equations on hyperbolic space

Abstract In this paper we study linear fractional differential equations involving tempered Caputo-type derivatives in the hyperbolic space. We consider in detail the three-dimensional case for its simple and useful structure. We also discuss the probabilistic meaning of our results in relation to the distribution of an hyperbolic Brownian motion time-changed with the inverse of a tempered stable subordinator. The generalization to an arbitrary dimension n can be easily obtained. We also show that it is possible to construct a particular solution for the non-linear porous-medium type tempered equation by using elementary functions.

Two fast finite difference methods for a class of variable-coefficient fractional diffusion equations with time delay

This paper introduces the fast Crank–Nicolson (CN) and compact difference schemes for solving the one- and two-dimensional fractional diffusion equations with time delay. The CN method is employed for temporal discretization, while the fractional centered difference (FCD) formula discretizes the Riesz space derivative. Additionally, a novel fourth-order scheme is developed using compact difference operators to improve accuracy. The convergence and stability of these schemes are rigorously proven. The discretized systems combining Toeplitz-like structures can be effectively solved by Krylov subspace solvers with suitable preconditioners. Each time level of these methods requires a computational complexity of O(plogp) per iteration and a memory of O(p), where p represents the total number of grid points in space. Numerical examples are given to illustrate both the theoretical results and the computational efficiency of the fast algorithm.

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

Shifted Legendre Gauss‐Lobatto collocation scheme for solving nonlinear coupled space‐time fractional reaction‐advection diffusion equations

AbstractThis work aims to develop and apply an algorithm for solving space‐time fractional reaction‐advection diffusion (STFRAD) nonlinear coupled equations with specified initial and boundary conditions by employing a Shifted Legendre Gauss‐Lobatto collocation (SLGLC) scheme. The fractional derivatives of time and space are defined in Caputo sense. The approximate solutions have been constructed with the help of shifted Legendre polynomials. The proposed numerical scheme reduces the coupled fractional order differential equations into a system of algebraic equations which can then be easily solved. The error analysis through the two test problems is carried out to validate the efficiency and efficacy of the method. The effect of advection and reaction terms, and various space‐time fractional order derivatives on the solution profiles have been studied and are shown graphically for different particular cases. It has been observed that the species whose transport is simulated using STFRADE shows higher concentration at a specific distance in porous media in case of integer order differential equations as compared to fractional order equations demonstrating overestimation of its impact in case of integer order differential equation. This is extremely important observation with respect to applications in the field of groundwater pollution management. The salient feature of the article is the stability and convergence analyses of the proposed scheme on the concerned model, which demonstrates the effectiveness and capabilities of the developed method.

A variable‐step high‐order scheme for time‐fractional advection‐diffusion equation with mixed derivatives

AbstractWe consider a high accuracy numerical scheme for solving the two‐dimensional time‐fractional advection‐diffusion equation including mixed derivatives, where the variable‐step Alikhanov formula and a fourth‐order compact approximation are employed to time and space derivatives, respectively. Under mild assumptions on the time step‐sizes, we obtain the unconditional stability and high‐order convergence (second‐order in time and fourth‐order in space) of the proposed scheme by energy method. The theoretical statements are justified by the numerical experiments.

New exact solutions of the conformable space-time two-mode foam drainage equation by two effective methods

In this article, the two-mode foam drainage equation in terms of time and space conformable sense has been investigated. Two effective methods, the generalized exponential rational function method (GERFM) and the improved version of the Bernoulli sub-equation function method (IBSEFM), are used to get new solutions of underlying equation. The fractional travelling wave transformation is applied to convert nonlinear partial differential equations to nonlinear ordinary differential equations. Proposed methods successfully extract trigonometric, hyperbolic and exponential solutions. Some of the obtained solutions are visualized to understand the effect of fractional orders of time and space derivatives on the wave profile and the dynamic behavior of the solutions.

Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

The influence of metal-free thiophene spacer chain on optoelectronic analysis by TD-DFT method for efficient dye-sensitized solar cells with enhanced non-linear optical activity

The pyrene group in conjugation with the thiophene moiety was used to design new organic molecules for the dye-sensitized solar cells (DSSCs) along with non-linear optical (NLO) properties. Five new thiophene chain conjugation molecules (PTRA-SPn, n = 1–5) were formed based on the (E)-2-(5-oxo-4-((5-(pyren-1-yl)thiophen-2-yl)methylene)-2-thioxothiazolidin-3-yl)acetic acid (PTRA). To comprehend the function of conjugation in varying the optoelectronic characteristics of these compounds, the π-spacer chain has been screened. The techniques of density functional theory (DFT) and time-dependent DFT (TD-DFT) have been used to study the electronic structures, Ultraviolet–Visible (UV–Vis) absorptions, and NLO activities of donor-spacer-acceptor (D-π-A) structures. According to the computed results, PTRA-SP5 exhibits a smaller energy gap (Eg) of 2.11 eV. The maximum UV–Vis spectrum has been found in PTRA-SP5 with a wavelength (λmax) of 532 nm. The reorganizational energy (RE) and dye regeneration (ΔGreg) of PTRA-SP5 are the smallest (λ- = 0.2340 eV, λ+ = 0.2122 eV, ΔGreg = 0.21 eV) compared to other sensitizers. The lowest exciton binding energies (Eb) of PTRA-SP2 (0.06 eV) and PTRA-SP3 (0.06 eV) are the lowest ones. The open circuit photovoltage (eVoc) for PTRA-SP5 (1.10 eV) is the highest. The NLO activity of the dipole moment (μnormal) and first-order hyperpolarizability (β) values of PTRA-SP5 achieve higher values (9.00 Debye and 34.82 × 10−30 esu, respectively). In summary, this work refined the impact of chain π-linkers on the photophysical characteristics of D-π-A spacer derivatives and offered some helpful suggestions to create new organic materials for optoelectronic uses.

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the L2 norm, the accuracy order, and the CPU time.

Generalized fractional bi-Hamiltonian structure of Plebański’s second heavenly equation in terms of conformable fractional derivatives

In this paper, we propose a symplectic and bi-Hamiltonian structure to a generalized (3+1) -dimensional Plebański’s second heavenly (SH) equation by using a conformable fractional derivative. We write the SH equation in terms of conformable fractional derivatives by replacing classical time (t) and space (z) derivatives with conformable fractional derivatives. We name this equation as a conformable fractional second heavenly (CFSH) equation and cast it in a two-component evolutionary form. We construct a degenerate Lagrangian for a two-component conformable fractional system and find a new Euler–Lagrange formalism for four-dimensional Lagrange density. We apply Dirac’s theory of constraints to the CFSH equation and we prove that it admits bi-Hamiltonian structure according to Magri’s theorem.

High-order second-generation wavelet upwind schemes with multiresolution self-adaptive capabilities for hyperbolic conservation laws

We introduce a novel system of high-order wavelet collocation upwind schemes utilizing a broad class of second-generation wavelets for hyperbolic conservation laws. These schemes possess multiresolution self-adaptive capabilities and are integrated into a meshfree framework. Unlike traditional high-order schemes that necessitate frequent variable transformations between physical and characteristic spaces involving numerous local projections, our schemes enable the direct discretization of relevant spatial derivatives in the physical space. The newly proposed schemes make the most use of wavelet properties, i.e., applying asymmetrical interpolating wavelets to achieve the upwind property and utilizing the symmetrical second-generation wavelets to preserve shapes of functions and obtain higher data compression rates. Numerical tests of which solutions contain strong discontinuities and different scale smooth structures are conducted to demonstrate the enhanced performance of the proposed schemes based on the second-generation wavelets. The proposed schemes can utilize only 9 % of the nodes in the WENO5 scheme to yield similar results when addressing the advection of a square wave and achieve the convergent solution for the two interacting blast waves problem without local characteristics projection. In comparison to conventional methods and wavelet upwind schemes based on the interpolating wavelets we previously introduced, the newly proposed schemes exhibit a significantly higher data compression rate and yield substantial computational savings without compromising accuracy.

An efficient spline-based DQ method for 2D/3D Riesz space-fractional convection–diffusion equations

This paper proposes an efficient spline-based DQ method for the 2D and 3D convection–diffusion equations (CDEs) with Riesz fractional derivative in space, which have been widely used to describe the anomalous solute transport in complex media. Firstly, a spline-based differential quadrature (DQ) formula is developed to approximate the Riesz derivative by using cubic B-splines as trial functions, which allows us to approximate the fractional derivatives with high accuracy and small computational cost. We then utilize it to discretize the fractional derivatives in the governing equation and a cubic B-spline DQ scheme is further established by applying the finite difference (FD) scheme to the resulting system of ordinary differential equations. A brief implementation of the proposed DQ method is also presented. To examine the effectiveness of this spline-based DQ method, numerical tests are finally done on some benchmark problems and the simulation of rotating Gaussian hill in convection-dominated flow governed by fractional derivatives. The advantages in computational accuracy and efficiency are illustrated by comparing the results with the other algorithms in open literature.

An efficient numerical approach for singularly perturbed time delayed parabolic problems with two-parameters

ObjectivesThe main objective of this work is to design an efficient numerical scheme is proposed for solving singularly perturbed time delayed parabolic problems with two parameters.ResultsThe scheme is constructed via the implicit Euler and non-standard finite difference method to approximate the time and space derivatives, respectively. Besides, to enhance the accuracy and order of convergence of the method Richardson extrapolation technique is employed. Intensive numerical experimentation has been done on some model examples. Further, the layer behavior of the solutions is presented using graphs and observed to agree with the existing theories. Finally, error analysis of the scheme is done and observed that the proposed method is parameter uniform convergent with the order of convergence (Δt)2+h2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( (\\Delta t )^2+h^2\\right) $$\\end{document}

Fundamental solution of the time-space bi-fractional diffusion equation with a kinetic source term for anomalous transport

The purpose of this paper is to study the fundamental solution of the time-space bi-fractional diffusion equation incorporating an additional kinetic source term in semi-infinite space. The equation is a generalization of the integer-order model ∂tρ(x,t)=∂x2ρ(x,t)-ρ(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _{{t}} {\\rho (x,t)}} = {\\partial ^2_{{x}} {\\rho (x,t)}} - { \\rho (x,t)}$$\\end{document} (also known as the Debye–Falkenhagen equation) by replacing the first-order time derivative with the Caputo fractional derivative of order 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\alpha < 1$$\\end{document}, and the second-order space derivative with the Riesz-Feller fractional derivative of order 0<β<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< \\beta <2$$\\end{document}. Using the Laplace-Fourier transforms method, it is shown that the parametric solutions are expressed in terms of the Fox’s H-function that we evaluate for different values of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}.

Stiff-cut leap-frog scheme for fractional Laplacian diffusion equations

In this study, we investigate the numerical discretization of two-dimensional fractional Laplacian diffusion equations. After discretizing the space derivative, we split the discrete fractional Laplacian into two parts. One is the τ preconditioner multiplying a constant, solved using the leap-frog scheme, while the other part and the remaining nonlinear term are treated with an explicit scheme. This results in what is termed the stiff-cut leap-frog scheme, which is more efficient than the traditional leap-frog scheme. The main advantage of the proposed scheme is that the coefficient matrices of the generated linear systems belong to τ algebra, which can be diagonalized by fast discrete sine transform with the complexity of O(NlogN), where N is the freedom of unknowns. Hence, no iterative method is required at each time integration. Moreover, we theoretically prove that the proposed scheme is unconditionally stable and converges in second order. Ultimately, numerical experiments are conducted to demonstrate the precision and effectiveness of the stiff-cut leap-frog scheme.

Haar wavelet collocation technique for numerical solution of porous media equations

This paper focuses on using efficient Haar wavelet collocation method to nonlinear parabolic differential equations in order to solve steady state pore-scale creeping fluid flow in porous medium, substituting finite Haar series for space derivatives, incontrast forward difference approach is used to estimate the time derivative for porous media equation. These equations has vast number of application in engineering and various other fields such as, in petroleum engineering, there are often situations where fluids are near critical conditions. As oil pressure reduces below the bubble point when gas forms liquid condensations, or when the surface tension become very low i.e, transition from miscibility to immiscibility. To model gas production by pressure decline requires an accurate description of physical mechanism involved in the appearance of the gas bubbles. The observed stability with theoretical convergence assertion, the conservation law of mass and energy are stated. This supports the convergence of the preferred paradigm. A convergent approximation to the porous media equation is obtained by suggested approach. By calculating the maximum error norm and the experimental rate of convergence for various problems, the method efficiency and dependability are demonstrated. The method is accurate, easily applicable, and efficient.

Global existence and blowup of smooth solutions to the semilinear wave equations in FLRW spacetime

We are interested in the semilinear wave equations evolving in the expanding spacetimes with Friedmann–Lemaître–Robertson–Walker (FLRW) metric. By the weighted energy estimate, we show that when the nonlinearity depends on the time derivative of the unknown, the equation admits a global smooth solution if the spacetime is undergoing accelerated expansion. While the solution will blowup in the sense of some averaged quantity if the expanding rate is not fast enough. When the nonlinearity depends on the space derivatives of the unknown or the unknown itself, we can show that the solution will blowup in finite time even though the expanding rate is fast enough (accelerated expansion). Our results show that the semilinear wave equations in FLRW spacetimes have different properties from the famous Glassey and Strauss conjectures in flat or asymptotically flat spacetimes.

Numerical solution of Fisher’s equation through the application of Haar wavelet collocation method

This research article presents algorithm that employs the Haar wavelet collocation method (HWCM) to solve the convection diffusion reaction (CDR) model, specifically Fisher’s equation (F.E.). The proposed method is suitable for addressing both linear and nonlinear problems. The Haar wavelet method is a numerical technique that utilizes wavelet basis functions to provide an approximation of the solution for the Fisher’s equations. Additionally, a forward difference scheme is utilized to approximate the time derivative while the highest order space derivative is replaced by Haar function. The performance of the technique is demonstrated through some examples investigated in this study. The maximum absolute errors are computed for varying numbers of collocation points. The HWCM provides superior accuracy, flexibility, convenience, and computational efficiency compared to alternative methods.