This study presents an enhanced version of the Temimi-Ansari Method (TAM) for effectively solving nonlinear integro-differential equations involving Fredholm-type integrals. The improved method builds upon the original TAM framework and …
This study presents an enhanced version of the Temimi-Ansari Method (TAM) for effectively solving nonlinear integro-differential equations involving Fredholm-type integrals. The improved method builds upon the original TAM framework and demonstrates its robustness in addressing complex functional equations. Symbolic computation tools are employed to implement the method, and its performance is illustrated through several benchmark problems. The obtained results are compared with exact solutions and other semi-analytical techniques to validate the accuracy and efficiency of the proposed approach. The method proves to be computationally efficient, capable of simplifying calculations, and suitable for solving both linear and nonlinear Fredholm integro-differential equations of the second kind.
This study introduces a reliable semi-analytical approach for solving partial differential equations (PDEs) using a Modified New Iterative Method (MNIM). The primary aim is to enhance the efficiency of deriving …
This study introduces a reliable semi-analytical approach for solving partial differential equations (PDEs) using a Modified New Iterative Method (MNIM). The primary aim is to enhance the efficiency of deriving closed-form solutions through an innovative formulation of an integral operator based on n-fold integration. This approach circumvents the conventional necessity of transforming PDEs into systems of multiple integral equations, thereby streamlining the solution process. The effectiveness of the MNIM is assessed through a series of examples, demonstrating its rapid convergence and superior performance in solving an array of evolution and partial differential equations. The results indicate that the MNIM not only simplifies the solution process but also significantly improves computational efficiency compared to traditional methods. This contribution holds substantial implications for both theoretical advancements in numerical analysis and practical applications across various fields where PDEs are prevalent, thereby facilitating more effective problem-solving strategies in complex systems.
In this study, we propose a novel hybrid analytical technique that combines the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve a class of linear and …
In this study, we propose a novel hybrid analytical technique that combines the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve a class of linear and nonlinear first-order initial value problems (IVPs), including those of fractional order. The principal aim of this approach is to overcome the computational challenges typically encountered in each individual method—namely, the complexity of generating Adomian polynomials in ADM and the requirement for Lagrange multipliers in VIM. By synthesizing the strengths of both methods, the hybrid scheme constructs analytical series solutions without necessitating linearization, Adomian polynomials, or the explicit formulation of Lagrange multipliers. This significantly streamlines the solution process while preserving accuracy and generality. The validity and computational efficiency of the proposed method are substantiated through a series of illustrative examples, encompassing both integer-order and fractional differential equations. The results demonstrate that the hybrid approach not only simplifies implementation but also yields precise and rapidly converging solutions, making it a robust alternative for tackling a broad spectrum of initial value problems in mathematical modeling and applied sciences.
In this study, we propose a novel hybrid analytical technique that combines the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve a class of linear and …
In this study, we propose a novel hybrid analytical technique that combines the Adomian Decomposition Method (ADM) with the Variational Iteration Method (VIM) to solve a class of linear and nonlinear first-order initial value problems (IVPs), including those of fractional order. The principal aim of this approach is to overcome the computational challenges typically encountered in each individual method—namely, the complexity of generating Adomian polynomials in ADM and the requirement for Lagrange multipliers in VIM. By synthesizing the strengths of both methods, the hybrid scheme constructs analytical series solutions without necessitating linearization, Adomian polynomials, or the explicit formulation of Lagrange multipliers. This significantly streamlines the solution process while preserving accuracy and generality. The validity and computational efficiency of the proposed method are substantiated through a series of illustrative examples, encompassing both integer-order and fractional differential equations. The results demonstrate that the hybrid approach not only simplifies implementation but also yields precise and rapidly converging solutions, making it a robust alternative for tackling a broad spectrum of initial value problems in mathematical modeling and applied sciences.
This study introduces a reliable semi-analytical approach for solving partial differential equations (PDEs) using a Modified New Iterative Method (MNIM). The primary aim is to enhance the efficiency of deriving …
This study introduces a reliable semi-analytical approach for solving partial differential equations (PDEs) using a Modified New Iterative Method (MNIM). The primary aim is to enhance the efficiency of deriving closed-form solutions through an innovative formulation of an integral operator based on n-fold integration. This approach circumvents the conventional necessity of transforming PDEs into systems of multiple integral equations, thereby streamlining the solution process. The effectiveness of the MNIM is assessed through a series of examples, demonstrating its rapid convergence and superior performance in solving an array of evolution and partial differential equations. The results indicate that the MNIM not only simplifies the solution process but also significantly improves computational efficiency compared to traditional methods. This contribution holds substantial implications for both theoretical advancements in numerical analysis and practical applications across various fields where PDEs are prevalent, thereby facilitating more effective problem-solving strategies in complex systems.
This study presents an enhanced version of the Temimi-Ansari Method (TAM) for effectively solving nonlinear integro-differential equations involving Fredholm-type integrals. The improved method builds upon the original TAM framework and …
This study presents an enhanced version of the Temimi-Ansari Method (TAM) for effectively solving nonlinear integro-differential equations involving Fredholm-type integrals. The improved method builds upon the original TAM framework and demonstrates its robustness in addressing complex functional equations. Symbolic computation tools are employed to implement the method, and its performance is illustrated through several benchmark problems. The obtained results are compared with exact solutions and other semi-analytical techniques to validate the accuracy and efficiency of the proposed approach. The method proves to be computationally efficient, capable of simplifying calculations, and suitable for solving both linear and nonlinear Fredholm integro-differential equations of the second kind.