The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.

The $\mathsf{q}$-deformed Sinh-Gordon equation extends the classical Sinh-Gordon equation by incorporating a deformation parameter $\mathsf{q}$. It provides a framework for studying nonlinear phenomena and soliton dynamics in the presence of quantum deformations, leading to intriguing mathematical structures and potential applications in diverse areas of physics. In this work, we imply the homotopy analysis method, to obtain approximate solutions for the proposed equation, the error estimated from the obtained solutions reflects the efficiency of the solving method. The solutions were presented in the form of 2D and 3D graphics. The graphics clarify the impact of a set of parameters on the solution, including the deformation parameter $\mathsf{q}$, as well as the effect of time and the fractional order derivative.

In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). The proposed scheme uses Shifted Chebyshev fifth-kind polynomials with the spectral collocation approach. Besides, the proposed GFPDEs represent a great generalization of significant types of fractional partial differential equations (FPDEs) and their applications, which contain many previous reports as a special case. The fractional differential derivatives are expressed in terms of the Caputo sense. Moreover, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of differential equations which can be solved numerically. In addition, the classical fourth-order Runge-Kutta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple, easy to implement and effective numerical method.

This paper presents an innovative approach to solve $\mathit{q}$-fractional partial differential equations through a combination of two semi-analytical techniques: The Residual Power Series Method (RPSM) and the Homotopy Analysis Method (HAM). Both methods are extended to obtain approximations for $\mathit{q}$-fractional partial differential equations ( $\mathit{q}$-FPDEs). These equations are significant in $\mathit{q}$-calculus, which has gained attention due to its relevance in engineering applications, particularly in quantum mechanics. In this study, we solve linear and nonlinear $\mathit{q}$-FPDEs and obtain the closed-form solutions, which confirm the validity of the utilized methods. The results are further illustrated through two-dimensional and three-dimensional graphs, thus highlighting the interaction between parameters, particularly the fractional parameter, the $\mathit{q}$-calculus parameter, and time.

This paper presented the formulation and solution of the time fractional q-deformed tanh-Gordon equation, a new extension to the traditional tanh-Gordon equation using fractional calculus, and a q-deformation parameter. This extension aimed to better model physical systems with violated symmetries. The approach taken involved the controlled Picard method combined with the Laplace transform technique and the Caputo fractional derivative to find solutions to this equation. Our results indicated that the method was effective and highlighted our approach in addressing this equation. We explored both the existence and the uniqueness of the solution, and included various 2D and 3D graphs to illustrate how different parameters affect the solution's behavior. This work aimed to contribute to the theoretical framework of mathematical physics and has potential applications across multiple interdisciplinary fields.

This paper presents a significant contribution in the form of a new general equation, namely the $\mathfrak{q}$-deformed equation or the $\mathfrak{q}$-deformed tanh-Gordon equation. The introduction of this novel equation opens up new possibilities for modeling physical systems that exhibit violated symmetries. By employing the $(G^{\prime }/G)$ expansion method, we have successfully derived solitary wave solutions for the newly defined $\mathfrak{q}$-deformed equation under specific parameter regimes. These solutions provide valuable insights into the behavior of the system and its dynamics. To further validate the obtained analytical results, the numerical solution of the $\mathfrak{q}$-deformed equation has been constructed by using the finite difference method. This numerical approach ensures the accuracy and reliability of the findings. To facilitate a comprehensive understanding of the results, we have included two- and three-dimensional tables and figures, which provide visual representations and comparisons between the analytical and numerical solutions. These graphical illustrations enhance the clarity and interpretation of the obtained data. The significance of the $\mathfrak{q}$-deformation lies in its ability to model physical systems that exhibit deviations from standard symmetry properties, such as extensivity. This type of modeling is increasingly relevant in various fields, as it allows for a more accurate representation of real-world phenomena.

This paper presents a comprehensive study of the (2+1) time-fractional nonlinear generalized biological population model (TFNBPM) using the $J$-transform combined with the optimal homotopy analysis method, a robust semi-analytical technique. The primary focus is to derive analytical solutions for the model and provide a thorough investigation of the convergence properties of these solutions. The proposed method allows for flexibility and accuracy in handling nonlinear fractional differential equations (NFDEs), demonstrating its efficacy through a series of detailed analyses. To validate the results, we present a set of 2D and 3D graphical representations of the solutions, illustrating the dynamic behavior of the biological population over time and space. These visualizations provide insightful perspectives on the population dynamics governed by the model. Additionally, a comparative study is conducted, where our results are juxtaposed with those obtained using other established techniques from the literature. The comparisons underscore the advantages of optimal homotopy analysis $J$-transform method (optimal HA $J$-TM), highlighting its consistency and superior convergence in solving complex fractional models.