Özet:
Linear and nonlinear fractional-order pseudo-hyperbolic telegraph partial differential equations have important applications in diverse fields including engineering, physics, and finance. However, analytical solutions to such equations are challenging to obtain. This thesis investigates accurate and efficient numerical techniques for solving linear and nonlinear fractional-order pseudo-hyperbolic telegraph partial differential equations defined by Caputo fractional derivatives. Initially, a modified double Laplace transform method is developed to get analytical solutions for linear fractional-order pseudo-hyperbolic telegraph partial differential equations. The method's simplicity and effectiveness are exemplified through a problem demonstration. In addition, this study presents a comprehensive methodology for identifying mild solutions to abstract form of nonlinear fractional-order pseudo-hyperbolic telegraph partial differential equations. The stability of these solutions is also investigated. The abstract technique is demonstrated to have broad applicability across many types of nonlinearity. Furthermore, a specific numerical technique called explicit finite difference approach is formulated and examined for solving linear fractional-order pseudo-hyperbolic telegraph partial differential equations. Stability estimates are calculated and the method is validated on an example problem, showing a high level of agreement between the numerical and precise solutions. Furthermore, a special application of an explicit finite difference technique is utilised to solve nonlinear fractional-order pseudo-hyperbolic telegraph partial differential equations. The implementation of this scheme is carried out using MATLAB. Numerical experiments demonstrate stability and convergence for different fractional orders, and effectively reflect the wave behaviour and memory effects. In summary, this thesis presents novel numerical and analytical approaches for solving linear and nonlinear fractional-order pseudo-hyperbolic telegraph partial differential equations. Both finite difference and modified double Laplace transform methods are developed and shown to be accurate, stable, and computationally efficient. These techniques will enable improved modeling and understanding of complex systems exhibiting fractional dynamics and memory effects across science and engineering disciplines.
