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ItemNUMERICAL APPROXIMATIONS FOR THE FRACTIONAL LAPLACIAN IN SPACE-FRACTIONAL REACTION-DIFFUSION EQUATIONS(Middle Tennessee State University, 2020) Alzahrani, Saham ; Khaliq, Abdul Q. M. ; Melnikov, Yuri ; Koritsanszky, Tibor ; Computational SciencesThe systems of non-linear time-dependent space-fractional differential equations have been employed to model important physical phenomena in many fields of engineering and science. The analytical solutions of most of these systems are unknown, and evaluating an analytical solution for some fractional differential equations is complicated and difficult to calculate because it is in the trigonometric series form. Thus, developing numerical solutions for such nonlinear systems is essential. There have been growing interests recently to develop efficient and robust numerical schemes for solving the nonlinear systems of fractional differential equations. In this study, several novel numerical schemes are proposed to solve the systems of multidimensional non-linear space-fractional reaction-diffusion equations efficiently. The non-local nature of the fractional operator adds new features to the mathematical models but also introduces additional difficulties in their implementation where large, dense matrices are required at each time step. To overcome this challenge, the Fourier spectral approach is applied to discretize the fractional Laplacian. This approach gives a diagonal representation of the fractional operator while achieving spectral convergence and the implementation to multi-dimensions is similar to onedimensional problems. Since this approach lacks capability to implement on nonhomogeneous boundary conditions, a second-order matrix transfer technique (MTT) for non-homogeneous boundary conditions is used for the space discretization. A fourth-order MTT based on a compact scheme is also employed for the discretization of the fractional Laplacian. To deal with the nonlinearities, exponential time differencing schemes (ETD) are employed for the reason that while the approaches achieve the expected accuracy, solving nonlinear systems at each time step is no longer needed. The Fourier spectral approach is combined with two second-order ETD schemes to solve space-fractional reaction-diffusion equations with non-smooth initial data and is also combined with a forth-order ETD scheme to provide highly efficient solutions for multidimensional v systems. The second-order MTT is combined with the forth-order ETD scheme to solve problems with non-homogeneous boundary conditions. Moreover, the fourthorder compact scheme MTT is combined with forth-order ETD schemes to show the effectiveness of the L-stable scheme when the initial data is non-smooth and to illustrate that the A-stable scheme is not reliable for some time steps. A novel reliability constraint is introduced to avoid the oscillations present in the solutions when the A-stable scheme is employed. Theoretical and numerical investigation of the convergence and stability of the numerical schemes have been discussed. Extensive numerical experiments are performed on wide well-known systems of time-dependent space-fractional reaction-diffusion equations to demonstrate the reliability, efficiency and accuracy of the developed schemes.
ItemSimulation of Multiple Inheritance in Java(Middle Tennessee State University, 2016) Swaroop, Sonal ; Computational SciencesMultiple inheritance is a cornerstone of OOPs paradigms with benefits such as reusability (using methods of parent class by child classes), extensibility (extending the parent class logic as per business logic of the child class), data hiding (base class may keep some data private so that it cannot be altered by the derived class), and lesser compilation time. But there is no consensus among researchers on the semantics of multiple inheritance in the presence of method overriding and potential conflicts due to multiple definitions. C++, Common lisp and few other languages supports multiple inheritance while java doesn't support it. It is just to remove ambiguity, because multiple inheritance can cause ambiguity in few scenarios such as Diamond problem. James Gosling quotes “JAVA omits many rarely used, poorly understood, confusing features of C++ that in our experience bring more grief than benefit. This primarily consists of operator overloading and multiple inheritance. Java creators provided users with much easier and robust alternative to multiple inheritance i.e. interfaces, composition and abstract classes that alleviate problem during casting and constructor chaining. The primary objective of this research study is to develop a trade-off between multiple inheritance in C++ and substitution for multiple inheritance in Java and conclude if multiple inheritance is good programming technique and if Java should introduce direct implementation of multiple inheritance. The secondary objective is to introduce novel ways to substitute multiple inheritance in Java such as twin pattern and dynamic multiple inheritance, and conclude if such implementations may bring much easier and feasible alternatives to interfaces