Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2 − 1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Acknowledgments: M.A.Z. wishes to acknowledge the support of Nazarbayev University Program 091019CRP2120 and the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”). M.A.Z. wishes also to acknowledge the financial support of the National Research Centre of Egypt (NRC)

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Numerical investigation of two models of nonlinear fractional reaction subdiffusion equations

We consider new numerical schemes to solve two different systems of nonlinear fractional reaction subdiffusion equations. These systems of equations model the reversible reaction A+B⇌C in the presence of anomalous subdiffusion. The first model is based on the Henry \& Wearne [1] model where the reaction term is added to the subdiffusion equation. The second model is based on the model by Angstmann, Donnelly \& Henry [2] which involves a modified fractional differential operator. For both models the Keller Box method [3] along with a modified L1 scheme (ML1), adapted from the Oldham and Spanier L1 scheme [4], are used to approximate the spatial and fractional derivatives respectively. Numerical prediction of both models were compared for a number of examples given the same initial and boundary conditions and the same anomalous exponents. From the results, we see similar short time behaviour for both models predicted. However for long times the solution of the second model remains positive whilst the Henry \& Wearne based–model predictions may become negative

On the numerical solution of the Lane-Emden, Bratu and Troesch equations.

Masters Degree. University of KwaZulu-Natal, Pietermaritzburg.Many engineering and physics problems are modelled using differential equations, which may be highly nonlinear and difficult to solve analytically. Numerical techniques are often used to obtain approximate solutions. In this study, we consider the solution of three nonlinear ordinary differential equations; namely, the initial value Lane-Emden equation, the boundary value Bratu equation, and the boundary value Troesch problem. For the Lane- Emden equation, a comparison is made between the accuracy of solutions using the finite difference method and the multi-domain spectral quasilinearization method along with the exact solution. We found that the multi-domain spectral quasilinearization method gave a better solution. For the Bratu problem, a comparison is made between the spectral quasilinearization method and the higher-order spectral quasilinearization method. The higher-order spectral quasilinearization method gave more accurate results. The Troesch problem is solved using the higher-order spectral quasilinearization method and the finite difference method. The solutions obtained are compared in terms of accuracy. Overall, the higher-order spectral quasilinearization method and multi-domain spectral quasilinearization method gave the accurate solutions, making these two methods to be the most reliable for these three problems

Numerical solution methods for fractional partial differential equations

Fractional partial differential equations have been developed in many different fields such as physics, finance, fluid mechanics, viscoelasticity, engineering and biology. These models are used to describe anomalous diffusion. The main feature of these equations is their nonlocal property, due to the fractional derivative, which makes their solution challenging. However, analytic solutions of the fractional partial differential equations either do not exist or involve special functions, such as the Fox (H-function) function (Mathai & Saxena 1978) and the Mittag-Leffler function (Podlubny 1998) which are diffcult to evaluate. Consequently, numerical techniques are required to find the solution of fractional partial differential equations. This thesis can be considered as two parts, the first part considers the approximation of the Riemann-Liouville fractional derivative and the second part develops numerical techniques for the solution of linear and nonlinear fractional partial differential equations where the fractional derivative is defied as a Riemann-Liouville derivative. In the first part we modify the L1 scheme, developed initially by Oldham & Spanier (1974), to develop the three schemes which will be defined as the C1, C2 and C3 schemes. The accuracy of each method is considered. Then the memory effect of the fractional derivative due to nonlocal property is discussed. Methods of reduction of the computation L1 scheme are proposed using regression approximations. In the second part of this study, we consider numerical solution schemes for linear fractional partial differential equations. Here the numerical approximation schemes are developed using an approximation of the fractional derivative and a spatial discretization scheme. In this thesis the L1, C1, C2, C3 fractional derivative approximation schemes, developed in the first part of the thesis, are used in conjunction with either the Centred-finite difference scheme, the Dufort-Frankel scheme or the Keller Box scheme. The stability of these numerical schemes are investigated via the technique of the Fourier analysis (Von Neumann stability analysis). The convergence of each the numerical schemes is also discussed. Numerical tests were used to conform the accuracy and stability of each proposed method. In the last part of the thesis numerical schemes are developed to handle nonlinear partial differential equations and systems of nonlinear fractional partial differential equations. We considered two models of a reversible reaction in the presence of anomalous subdiffusion. The Centred-finite difference scheme and the Keller Box methods are used to spatially discretise the spatial domain in these schemes. Here the L1 scheme and a modification of the L1 scheme are used to approximate the fractional derivative. The accuracy of the methods are discussed and the convergence of the scheme are demonstrated by numerical experiments. We also give numerical examples to illustrate the e�ciency of the proposed scheme