This study presented a novel approach to investigating the existence, uniqueness, and stability of solutions for an initial value problem involving fractional differential equations of variable order. In contrast to conventional methods in the literature, which often utilized generalized intervals and piecewise constant functions, we introduced a new fractional operator that is more appropriate for this problem. The existence and uniqueness of the solutions ware demonstrated through Leray-Schauder fixed point theorem and Banach's theorem, with an analysis of the uniform stability of the problem. The strength of our approach lies in its straightforwardness and reliance on fewer restrictive assumptions. The study concluded with an application that features a practical example, accompanied by visual illustrations.

In this article, a hybrid numerical scheme based on Lucas and Fibonacci polynomials in combination with Störmer's method for the solution of Klein/Sinh-Gordon equations is proposed. Initially, the problem is transformed to a time-discrete form by using Störmer's technique. Then, with the help of Fibonacci polynomials, we approximate the derivatives of the function. The suggested technique is validated to both one and two-dimensional problems. The resultant findings are compared with existing numerical solutions and presented in a tabular form. The comparison reveals the superior accuracy of the scheme. The numerical convergence of the scheme is computed in each example.

This study presents a numerical solution for the two-asset time-fractional Black-Scholes model, which governs American and digital options, using a local meshless collocation method based on Gaussian radial basis functions. The proposed meshless approach effectively discretized the spatial derivatives of the model, while the Caputo derivative was employed to represent the time-fractional component, capturing the memory effects and non-local properties characteristic of fractional-order models. Numerical assessments were conducted to evaluate the method's performance across these option models. The study discusses the handling of interest rates, highlighting the method's capability to manage the complexities inherent in multi-asset options. The efficacy and accuracy of the proposed meshless approach were evaluated using the $L_{\mathrm{\infty }}$ error norms. In the absence of exact solutions for these option models, the double mesh technique was utilized to validate the accuracy and efficiency of the proposed method, ensuring the robustness and reliability of the numerical results.

The infection caused by Rift Valley fever (RVF) virus is a dangerous vector-borne disease found in humans, domestic, and wild animals. It is transferred through insect vectors to ruminant host and then spread through direct contact of infected animals with their body fluid or organs. In this paper, a fractal-fractional model for the transmission of RVF in the Caputo's sense was presented. We analyzed the model and determined the basic reproduction number through the next-generation matrix technique, indicated by ${\mathcal{R}}_0$. The global sensitivity technique is used for the sensitivity test of ${\mathcal{R}}_0$ to find out the most sensitive input-factors to the reproduction parameter ${\mathcal{R}}_0$. The existence and uniqueness results of the proposed fractal-fractional model were established. Then, we presented the fractal-fractional dynamics of the proposed RVF model through a novel numerical scheme under the fractal-fractional Caputo operator. In the end, the recommended model of RVF was highlighted numerically with the variation of different input parameters of the system. The key factors of the system were highlighted to the policymakers for the control and prevention of the infection.

In this study, we present two meshless schemes, namely the radial basis function (RBF) method and the polynomial method, for the numerical investigation of the time-fractional Harry Dym equation and the Drinfeld-Sokolov-Wilson system. In both methods, the temporal derivatives are estimated using the Caputo operator, while the spatial derivatives are approximated either through radial basis functions or polynomials. Additionally, a collocation approach is employed to convert the system of equations into a system of linear equations that is easier to solve. The accuracy of the methods is assessed by calculating the $L_{\mathrm{\infty }}$ error norm, and the outcomes are displayed through tables and figures. The simulation results indicate that both methods exhibit strong performance in handling the fractional partial differential equations (PDEs) under investigation.