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 non...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.展开更多
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 develo...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.展开更多
文摘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.
文摘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.
