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In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of orders θ 1  and  θ 2 and Riemann–Liouville derivative of orders ϱ 1  and  ϱ 2 with the p -Laplacian operator, where n − 1 < θ 1 , θ 2 , ϱ 1 , ϱ 2 ≤ n , and n ≥ 3 . With the help of two Green’s functions G ϱ 1 w , ℑ , G ϱ 2 w , ℑ , the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.

In this article, we take a time-space fractional convection-diffusion problem with a nonlinear reaction term on a finite domain. We use the L1 operator to discretize the Caputo fractional derivative and the weighted shifted Grünwald difference(WSGD) method to approximate the Riesz fractional derivative. Furthermore, we apply the Crank Nicolson difference scheme with weighted shifted Grünwald-Letnikov and obtain that the numerical method is unconditionally stable and convergent with the accuracy of O(τ2-α + h2), where α ∈ (0,1]. For finding the numerical solution of the nonlinear system of equation, we apply the fixed iteration method. At the end, numerical simulations are treated to verify the effectiveness and consistency of the proposed method.

In this paper, we study the nonlinear Riesz space-fractional convection–diffusion equation over a finite domain in two dimensions with a reaction term. The Crank–Nicolson difference method for the temporal and the weighted–shifted Grünwald–Letnikov difference method for the spatial discretization are proposed to achieve a second-order convergence in time and space. The D’Yakonov alternating–direction implicit technique, which is effective in two–dimensional problems, is applied to find the solution alternatively and reduce the computational cost. The unconditional stability and convergence analyses are proved theoretically. Numerical experiments with their known exact solutions are conducted to illustrate our theoretical investigation. The numerical results perfectly confirm the effectiveness and computational accuracy of the proposed method.