This paper is concerned with the concept of almost (τ1, τ2)-continuous functions. Moreover, some characterizations of almost (τ1, τ2)-continuous functions are investigated.

Our main purpose is to introduce the concepts of upper and lower $s\beta(\star)$-continuous multifunctions. In particular, some characterizations of upper and lower $s\beta(\star)$-continuous multifunctions are investigated. Moreover, the relationships between $s\beta(\star)$-continuous multifunctions and almost $s\beta(\star)$-continuous multifunctions are discussed.

Our main purpose is to introduce the notion of weakly quasi $(\tau_1,\tau_2)$-continuous multifunctions.Furthermore, several characterizations of almost quasi $(\tau_1,\tau_2)$-continuous multifunctions are established.

This paper presents new concepts of continuous multifunctions defined from an ideal topological space into a bitopological space, called upper weakly $\tau^\star(\sigma_1,\sigma_2)$-continuous multifunctions and lower weakly $\tau^\star(\sigma_1,\sigma_2)$-continuous multifunctions. Furthermore, several characterizations and some properties concerning upper weakly $\tau^\star(\sigma_1,\sigma_2)$-continuous multifunctions and lower weakly $\tau^\star(\sigma_1,\sigma_2)$-continuous multifunctions are investigated. Moreover, the relationships between almost $\tau^\star(\sigma_1,\sigma_2)$-continuity andweak $\tau^\star(\sigma_1,\sigma_2)$-continuity are established.

This paper presents new classes of multifunctions called upper quasi $\theta(\tau_1,\tau_2)$-continuousmultifunctions and lower quasi $\theta(\tau_1,\tau_2)$-continuous multifunctions. Furthermore, several characterizations and some properties concerning upper quasi $\theta(\tau_1,\tau_2)$-continuous multifunctions and lower quasi $\theta(\tau_1,\tau_2)$-continuous multifunctions are established.

The present study proposes an approximate analytical solution to a nonlinear system of time-fractional partial differential equations. The Sadik residual power series method, which integrates the two-part Sadik integral transform with the residual power series technique, is employed to solve the fractional differential equation in the Caputo sense. Nonlinear problems with known and unknown solutions are examined to demonstrate the capacity of the technique. Numerical simulations and 3D visualizations are conducted for various values of the fractional order to further understand the solution’s behavior. Additionally, the results are validated against exact solutions or existing methodologies to ensure their reliability and accuracy. A key advantage of the proposed method is its ability to generate results without the need for Adomian polynomials, perturbation techniques, discretization, or linearization, enabling a more efficient.