A combination of classical and shifted Jacobi polynomials for two-dimensional time-fractional diffusion-wave equations

Two-dimensional multi-term time-fractional diffusion-wave equations (TFDWEs) have numerous applications in fields such as material science, complex media, thermodynamics, heat conduction, quantum systems, finance, and economics. This study focuses on solving a specific type of 2D TFDWE by developing a pseudo-operational collocation scheme that utilizes three-variable Jacobi polynomials over the domain [−1,1]×[−1,1]×[0,1]. To achieve this, we construct three-variable Jacobi polynomials using classical orthogonal Jacobi polynomials Piσ,ς(x) and Pjθ,ϑ(y) for x,y∈[−1,1], where the parameters σ,ς,θ, and ϑ are greater than −1. Additionally, we incorporate shifted orthogonal Jacobi polynomials Pkϵ,ɛ(t) for t∈[0,1] with ϵ,ɛ>−1. This represents the first instance of such a construction. Variations in the parameters σ,ς,θ,ϑ,ϵ, and ɛ lead to the generation of different basis functions, each with distinct root distributions that serve as collocation points. Specifically, if σ>ς, the roots are more densely concentrated around x=−1; if σ<ς, the roots cluster around x=1; and if σ=ς, the roots are uniformly distributed across both ends. The next step involves deriving pseudo-operational matrices of the integration for both fractional and integer orders corresponding to the three-variable basis vector. To accomplish this, the Kronecker product of integral pseudo-operational matrices connected to one-variable basis vectors is utilized. A simpler system of algebraic equations is derived by substituting matrix relations and approximations into the given equation and collocating the resulting algebraic equation at the collocation nodes. This simplification facilitates finding an approximate result compared to directly solving the original equation. Before tackling the given problem, the existence and uniqueness of the 2D multi-term time-fractional diffusion-wave equations (MTFDWEs) are examined. Error bounds for the approximate solutions derived from the proposed method and their derivatives concerning the independent variables are evaluated in a Jacobi–weighted Sobolev space. This assessment confirms that selecting an appropriate number of basis functions yields an approximate solution with satisfactory accuracy. To exhibit the efficiency and effectiveness of the method, several 2D TFDWEs are solved that incorporate a diverse range of fractional-order terms.