A Comprehensive Scheme for Approximating Fractional 3D Partial Differential Equations in Fluid Systems

Manoj Kumar; Dr. Binny Gupta

doi:10.59828/pijms.v1i2.10

Authors

Manoj Kumar Scholar, Department of Mathematics, Sri Khushaldas University, Pilibanga, Hanumangarh Author
Dr. Binny Gupta Assistant Professor, Department of Mathematics, Sri Khushaldas University, Pilibanga, Hanumangarh, Raj. Author

DOI:

https://doi.org/10.59828/pijms.v1i2.10

Abstract

Fractional calculus has attracted significant attention as a powerful tool for modeling physical and engineering systems with memory and hereditary effects. Fractional-order derivatives offer more realistic descriptions of phenomena such as anomalous diffusion, viscoelastic behavior, and wave propagation. Recent advances in analytical and numerical techniques have enabled effective treatment of fractional partial differential equations (FPDEs). In this chapter, three semi-analytical methods—Adomian Decomposition Method (ADM), Variational Iteration Method (VIM), and New Iterative Method (NIM)—are applied to obtain approximate analytical solutions of three-dimensional time-fractional diffusion, telegraph, and wave equations. These methods avoid discretization and linearization, produce rapidly convergent series solutions, and are validated through illustrative examples, demonstrating their accuracy and applicability to multidimensional time-fractional models.

Keywords: Fractional calculus, Time-fractional partial differential equations, Adomian Decomposition Method (ADM), Variational Iteration Method (VIM), New Iterative Method (NIM), Diffusion equation, Telegraph equation, Wave equation, Semi-analytical methods.