https://dgaa.ubonab.ac.ir/ Journal of Differential Geometry, Applications and Aspects en daily 1 Sat, 01 Mar 2025 00:00:00 +0330 Sat, 01 Mar 2025 00:00:00 +0330 https://dgaa.ubonab.ac.ir/article_721505.html In this paper, the numerical solution of the initial-value problem involving the time-fractional diffusion problem in the Caputo sense can be express as a series of the shifted Vieta-Fibonacci polynomials with unknown coefficients. Next, by making use of the collocation points and the relations between their coefficients via the boundary conditions, the recent problem is reduced to a system of fractional ordinary differential equations (SFODEs) with initial conditions. Then, utilizing the residual power series method (RPSM) on SFODEs, the analytic approximate solution can be achieved. To illustrate the simplicity and accuracy of the proposed method, some numerical examples are considered. https://dgaa.ubonab.ac.ir/article_730883.html The theory of m-th root Finsler metrics has been applied to Ecology, Biology, Seismic Ray Theory, Gravitation, etc. It is regarded as a direct generalization of Riemannian metric in a sense, that is, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann-Finsler metric. In this paper, we will study the class of quintic (α, β)-metrics. We show that 3-th root(α, β)-metrics has a unbound Cartan torsion. Also, we focus on the class of 3-th root (α, β)-metrics. We will study the bound Cartan torsion for a 3-th, 4-th and 5-th root (α, β)-metrics.