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Comparing Solutions to the Nonlinear Dissipative Wave Equation

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Journal of Applied Mathematics and Physics 2022年第4期10卷 1281-1296页

作者： Zaki Mrzog Alaofi Talaat Sayed El-Danaf silvestru sever dragomir Department of Mathematics College of Science and Arts King Khalid University Muhayil Asir KSA Mathematics College of Engineering & Science Victoria University Melbourne Australia Department of Mathematics Faculty of Sciences and Arts Taibah University Medina KSA

In previous decades, many of the practical problems arising in scientific fields such as physics, engineering, and mathematics have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the dissipative wave equation, has been found to have a plethora of useful applications in different fields. A special class of solutions has been studied for the dissipative wave equation including exact solutions and approximate solutions. The aim of this article is to compare the non-polynomial spline method and the cubic B-spline method with the solution of a nonlinear dissipative wave equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

关键词： Dissipative Wave Equation Cubic B-Spline Non-Polynomial Spline Truncation Error Von Neumann Stability

Adaptive Boundary Control for the Dynamics of the Generalized Burgers-Huxley Equation

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Open Journal of Applied Sciences 2022年第8期12卷 1416-1438页

作者： Zaki Mrzog Alaofi Talaat Sayed Ali Faisal Abd Alaal silvestru sever dragomir Department of Mathematics College of Science and Arts King Khalid University Muhayil Asir Saudia Arabia Mathematics College of Engineering & Science Victoria University Melbourne Australia Department of Mathematics and Computer Science Faculty of Science Menoufia University Shebeen El-Kom Egypt Department of Mathematics Faculty of Science Damanhour University Damanhour Egypt

This paper deals with the boundary control problem of the unforced generalized Burgers-Huxley equation with high order nonlinearity when the spatial domain is [0, 1]. We show that this type of equations are globally exponential stable in L2 [0, 1] under zero Dirichlet boundary conditions. We use an adaptive nonlinear boundary controller to show the convergence of the solution to the trivial solution and to show that it achieves global asymptotic stability in time. We introduce numerical simulation for the controlled equation using the Adomian decomposition method (ADM) in order to illustrate the performance of the controller.

关键词： Adaptive Boundary Control Generalized Burgers-Huxley Equation Stability Adomian Decomposition Method

Computational Analysis for Solving the Linear Space-Fractional Telegraph Equation

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Open Journal of Modelling and Simulation 2022年第3期10卷 267-282页

作者： Zaki Mrzog Alaofi Talaat Sayed El-Danaf Adel Hadhoud silvestru sever dragomir Department of Mathematics College of Science and Arts King Khalid University Muhayil Asir Saudi Arabia Mathematics College of Engineering & Science Victoria University Melbourne Australia Department of Mathematics Faculty of Sciences and Arts Taibah University Medina Saudi Arabia Department of Mathematics and Computer Science Faculty of Science Menoufia University Shebeen El-Kom Egypt

Over the last few years, there has been a significant increase in attention paid to fractional differential equations, given their wide array of applications in the fields of physics and engineering. The recent development of using fractional telegraph equations as models in some fields (e.g., the thermal diffusion in fractal media) has heightened the importance of examining the method of solutions for such equations (both approximate and analytic). The present work is designed to serve as a valuable contribution to work in this field. The key objective of this work is to propose a general framework that can be used to guide quadratic spline functions in order to create a numerical method for obtaining an approximation solution using the linear space-fractional telegraph equation. Additionally, the Von Neumann method was employed to measure the stability of the analytical scheme, which showed that the proposed method is conditionally stable. What’s more, the proposal contains a numerical example that illustrates how the proposed method can be implemented practically, whilst the error estimates and numerical stability results are discussed in depth. The findings indicate that the proposed model is highly effective, convenient and accurate for solving the relevant problems and is suitable for use with approximate solutions acquired through the two-dimensional differential transform method that has been developed for linear partial differential equations with space- and time-fractional derivatives.

关键词： Fractional Differential Equations Quadratic Spline Functions Linear Space-Fractional Telegraph Equation Von Neumann Stability

Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation

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American Journal of Computational Mathematics 2021年第3期11卷 189-206页

作者： Zaki Mrzog Alaofi Talaat Sayed Ali Faisal Abd Alaal silvestru sever dragomir Department of Mathematics College of Science and Arts King Khalid University Muhayil Asir Saudia Arabia Mathematics College of Engineering & Science Victoria University Melbourne Australia Department of Mathematics Faculty of Sciences and Arts Taibah University Medina Saudi Arabia Department of Mathematics Faculty of Science Damanhour University Damanhour Egypt

In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary diff

关键词： Non-Polynomial Spline Third-Order Dispersive Partial Differential Equation Stability Convergent

FUNCTIONAL INEQUALITIES AND BEST APPROXIMATION

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Analysis in Theory and Applications 1998年第2期14卷 90-98页

作者： sever S dragomir Sizwe Mabizela University of Timisoara Ramainia University of Cape Town Republic of South Africa

We investigate the relationship between best approximations by elements of closed convex cones and the estimation of functionals on an inner product space (X,)in terms of the inner product on X.

关键词： over Th FUNCTIONAL INEQUALITIES AND BEST APPROXIMATION real