Younis Abid Sabawi

In this study, we investigate the (2+1)-dimensional Chaffee–Infante equation, a fundamental model describing gas diffusion dynamics in homogeneous media. To enhance the model’s accuracy in capturing real-world diffusion processes, we incorporate a conformable fractional derivative operator, reflecting recent advancements in fractional calculus. The bifurcation structures of the traveling wave solutions are analyzed using a planar dynamical system approach, revealing intricate phase portraits under various parameter settings. To obtain exact solutions, we apply the generalized improved G ′ / G method, demonstrating its effectiveness in handling nonlinear wave models. Additionally, numerical simulations, including 3D surface plots and 2D contour graphs, provide a visual representation of the wave dynamics, offering deeper insights into the system’s complex behavior. These findings contribute not only to the theoretical understanding of fractional differential equations but also to their applications in various branches of physics, particularly in modeling anomalous diffusion and nonlinear wave propagation.

The paper offers a numerical approximation for fractional ordinary differential equations in which the first and second-order are replaced by a fractional derivative. We have employed the cubic B-spline approach coupled with the finite element approach to sort out such problems. Also discussed exploring fractional differential equations by proving the existence and uniqueness of their solutions, along with the correctness of their error estimations. The error estimation is derived by the Aubin–Nitsche duality argument. The efficiency of all the proposed approaches is compared in several numerical examples in order to verify, demonstrate and display the applicability of the approach, as well as to compare the estimated outcomes with analytical result.

This study focuses on the Fredholm integro-differential equations and is frequently found in areas of applied mathematics, physics, and engineering fields. To address these systems and obtain exact of solutions under suitable conditions, we propose a novel analytical approach. We choose the Haar wavelet colocation method for processing due to its simplicity, effectiveness, and ability to handle non-smooth solutions. The integral terms in these equations are determined using the trapezoidal rule, which effectively strikes a balance between accuracy and computational efficiency. These results are compared with analytical solutions. The comparisons show that the suggested strategy yields highly accurate results and offers a solid framework for solving the Fredholm integro-differential equations.

This paper aims to develop an efficient and accurate numerical method for solving the third-order nonlinear Whitham–Broer–Kaup (WBK) shallow water model. To achieve this objective, we applied a cubic B-spline-based finite element method, which offers higher-order continuity and precise geometric representation. The study of this model is important due to its wide applications in fluid dynamics, plasma physics, and nonlinear optics. We established the existence and uniqueness of the numerical solutions and performed a convergence analysis for the semi-discrete Galerkin scheme. The system of ordinary differential equations produced by the application of this method is solved by the Crank–Nicolson method. To validate the current approach, three test problems are analyzed. The errors are measured by using the [ ] and [ ] norms. Numerical results demonstrate that the method provides highly accurate approximations, with close agreement between numerical and exact solutions, confirming the efficiency, stability, and reliability of the cubic B-spline finite element method for modeling complex nonlinear wave phenomena. In addition, we compared the present method with several published methods. The results show that the proposed method provides a better solution in terms of absolute error and computational efficiency.

This work elaborates and investigates an effective numerical method for solving nonlinear Allen–Cahn equation. In particular, we focused on a time discretization scheme using the Crank–Nicolson (CN) method combined with space discretization using the non-lumping Galerkin finite element method. Fully discrete solutions address nonlinear terms that lose their real characteristics. Our approach avoids the lumping or linearization of the CN scheme and hence holds the accuracy regarding the treatment of non-linearity of this method. In addition, a priori error estimates concerning the semi-discrete scheme in the L∞(L2)-norm are derived. In addition, we investigated the stability of the fully discrete scheme to ensure its robustness. Numerical results are presented for problems with three different boundary conditions, and it is seen that these results are consistent with the analytical solution and exhibit good convergence with high accuracy.

This study explores numerical solutions to nonlinear reaction-diffusion equations, with a focus on modeling concentration profiles in catalytic pellets, crucial for many chemical engineering applications. The problem is discretised and investigated in both the temporal and spatial domains using finite element method. A weak formulation is developed, and the existence and uniqueness of the finite element method solution are established. Stability and convergence of the numerical schemes are rigorously analyzed, and Crank–Nicolson-based discretization is implemented for enhanced accuracy. Numerical results illustrate the effectiveness of finite element method and finite difference method, with close agreement between the methods. This comparison highlights finite element method’s potential advantages in catalytic process modeling, showcasing its effectiveness in solving nonlinear reaction-diffusion PDEs

In this paper, a numerical investigation of a class of parabolic Volterra integro-differential equations (parabolic-VIDEs) is conducted. The approach focuses on semi-discretizing parabolic-VIDEs by utilizing a second-order compact finite difference method O ( δ t 2 ) for the time variable and approximating the integral term using the trapezoidal rule. Further, the spatial derivative is approximated by Haar wavelet method. This hybrid methodology leverages the strengths of both techniques to solve the equations more efficiently and accurately. The error analysis, using L ₂ and L _∞-norms, shows low computational costs. Numerical experiments show that second-order accuracy in both time and space, respectively. Stability and convergence of the proposed method are given through Sobolve space. Some numerical examples are tested for the effectiveness and accuracy of the proposed method. Furthermore, the presented numerical approach is compared with standard finite difference numerical methods and the results are reported in a table.

This paper introduces a novel quartic B-spline collocation method to address the coupled Whitham–Broer–Kaup (WBK) problem. The WBK problem is a topic of interest in the study of nonlinear wave phenomena and has applications in various fields, including fluid dynamics, plasma physics, and nonlinear optics. The method combines spatial quartic B-spline scheme discretization, and Crank–Nicolson temporal discretization. It is unconditionally stable as proven by the Von-Neumann technique. Numerical examples demonstrate the method’s superior accuracy compared to existing solutions. Error analysis employs l 2 and l ∞ norms, while the method exhibits high computational efficiency. The nonlinearity is managed through Rubin-Graves linearization. Comparisons with prior approaches highlight its efficiency, stability, adaptability to complex problems. The quartic B-spline method is well-suited for simulating fluid flow phenomena in shallow water scenarios, offering high accuracy and low computational cost.