Paper Title
Iterative and Semi-Analytical Methods for Nonlinear Differential Equations: A Review with Emphasis on the Daftardar-Gejji–Jafari Method
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Registration ID: IJNRD_328903
Published ID: IJNRD2610093
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Keywords
nonlinear differential equations; Daftardar-Gejji–Jafari method; new iterative method; Adomian decomposition; homotopy methods; Fisher equation; Newell–Whitehead–Segel equation; Boussinesq equations; stiff systems; chaotic systems
Abstract
Nonlinear ordinary and partial differential equations rarely admit closed-form solutions, and their study has therefore passed from analytical solvability to approximation with controlled error. This article reviews the principal families of methods for nonlinear differential equations: classical root-finders and time integrators, iterative and semi-analytical schemes (Picard, Adomian decomposition, homotopy perturbation, variational iteration and homotopy analysis), and discretisation-based PDE solvers (finite difference, finite element, finite volume, spectral and multigrid). Particular attention is given to the Daftardar-Gejji–Jafari method (DJM), also called the New Iterative Method, which generates a convergent series without Adomian polynomials, embedding parameters or Lagrange multipliers. We survey its formulation, convergence theory and applications, the modified and hybrid schemes built around it, and the treatment of three benchmark problems (the Fisher, Newell–Whitehead–Segel and Boussinesq equations) and of stiff and chaotic systems. A comparative analysis shows that discretisation methods are theoretically mature but costly or geometrically restrictive, whereas semi-analytical series methods give closed-form solutions at low algebraic cost but rest on a thinner theory and have a limited domain of convergence. Five research gaps are identified.
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How To Cite (APA)
Praveen Kumar Yadav, Dr. Mamta, & Dr.Farha Khan (October-2026). Iterative and Semi-Analytical Methods for Nonlinear Differential Equations: A Review with Emphasis on the Daftardar-Gejji–Jafari Method. INTERNATIONAL JOURNAL OF NOVEL RESEARCH AND DEVELOPMENT, 11(10), a861-a871. https://ijnrd.org/papers/IJNRD2610093.pdf
Issue
Volume 11 Issue 10, October-2026
Pages : a861-a871
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Paper Reg. ID: IJNRD_328903
Published Paper Id: IJNRD2610093
Research Area: Mathematics All
Author Type: Indian Author
Country: Bhopal, M.P., India
Published Paper PDF: https://ijnrd.org/papers/IJNRD2610093.pdf
Published Paper URL: https://ijnrd.org/viewpaperforall?paper=IJNRD2610093
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