SU\G(𝔸)/K·U
Sign Up for free Log In

A One-Step Modified New Iterative Method for Solving Partial Differential Equation

Authors

I. Abdulmalik, A. M. Kwami, J. O. Okai, Aminu Barde, Ogboche Abichele, Adejoh Jeremiah
Type: Article
Publication Date: 2025-04-28
Citations: 0
DOI: https://doi.org/10.58578/yasin.v5i3.5498

Abstract

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.

Topics

Locations

A Comparative Analysis of Two Semi Analytic Approaches in Solving Systems of First-Order Differential Equations

2024-06-30
Khadeejah James Audu , O.O. Babatunde
The resolution of systems of first-order ordinary differential equations (ODEs) stands as a pivotal pursuit with extensive implications across scientific and engineering domains. In tackling this fundamental task, this study … The resolution of systems of first-order ordinary differential equations (ODEs) stands as a pivotal pursuit with extensive implications across scientific and engineering domains. In tackling this fundamental task, this study undertakes a rigorous comparative assessment of two semi-analytic methodologies, the Variational Iterative Method (VIM) and the New Iterative Method (NIM). Motivated by the need to address a critical research gap, our investigation delves into these approaches' relative merits and demerits. Firstly, it conducts a comprehensive examination of VIM, a well-established method, juxtaposed with NIM, a relatively unexplored approach, to uncover their comparative strengths and limitations. Secondly, the study contributes to the existing knowledge in numerical methods for ODEs by shedding light on essential performance characteristics, including convergence properties, computational efficiency, and accuracy, across a diverse array of ODE systems. Through meticulous numerical experimentation, we not only reveal practical insights into the efficacy of VIM and NIM but also bridge a significant knowledge gap in the field of numerical ODE solvers. Our findings highlight VIM as the more effective method, thus advancing our understanding of semi-analytic approaches for solving ODE systems and furnishing valuable guidance for practitioners and researchers in selecting the most suitable method for their specific applications
View PDF Chat

AN IN-DEPTH STUDY ON THE APPLICATION OF THE DIFFERENTIAL TRANSFORM METHOD FOR EFFECTIVELY SOLVING VARIOUS-ORDER ORDINARY DIFFERENTIAL EQUATIONS

2025-05-20
Olanrewaju Thomas Olotu , Kafilat Adebimpe Salaudeen , Rotimi Oluwafemi Folaranmi +2 more
This study presents a comprehensive analysis of the Differential Transform Method (DTM) as an effective tool for solving ordinary differential equations (ODEs) of various orders. Emphasis is placed on the … This study presents a comprehensive analysis of the Differential Transform Method (DTM) as an effective tool for solving ordinary differential equations (ODEs) of various orders. Emphasis is placed on the method’s ability to handle both linear and nonlinear ODEs without the need for common simplification techniques such as linearization, discretization, or perturbation, which often introduce additional complexities or reduce accuracy. By systematically applying DTM to different classes of ODEs, the study highlights its versatility and accuracy in handling initial value problems across a range of complexities with the solution of the first, second, third and fourth-orders ODEs. Comparative analyses with analytical methods demonstrate the superiority of DTM in terms of computational efficiency and solution accuracy. Additionally, graphical representations of both the analytical solutions and the approximate results obtained using DTM were plotted across various orders to showcase the robustness of the DTM. This article also includes detailed examples illustrating the step-by-step application of DTM, providing insights into its potential for broader applications in engineering, physics, and applied mathematics. The increasing complexity of systems modeled by ODEs in scientific and engineering fields necessitates efficient and accurate methods for obtaining reliable solutions, thereby justifying the need to explore alternative approaches like DTM. The findings underscore the relevance of DTM as a powerful method for solving ODEs of various orders, making it a valuableaddition to the toolbox of researchers and practitioners in the field. Conclusively, the results show that DTM is an efficient, accurate, and reliable method.

Introduction to Advanced Numerical Methods for Solving ODEs

2020-08-31
Narsimha Reddy , Syed Shahnawaz Ali , K. Narayan
This paper presents an in-depth exploration of advanced numerical methods for solving ordinary differential equations (ODEs), essential for modeling and understanding complex physical systems. Traditional methods often fall short in … This paper presents an in-depth exploration of advanced numerical methods for solving ordinary differential equations (ODEs), essential for modeling and understanding complex physical systems. Traditional methods often fall short in terms of accuracy and efficiency when applied to non-linear or stiff ODEs, necessitating the development of more sophisticated techniques. This study focuses on several advanced methods, including Runge-Kutta methods, multistep methods, and finite element methods, detailing their theoretical foundations and practical applications. Comparative analyses are provided to highlight the strengths and limitations of each approach, supported by numerical experiments and error analysis. The implementation challenges and computational aspects are also discussed, offering insights into the choice of appropriate methods for different types of ODE problems. This work aims to serve as a comprehensive guide for researchers and practitioners in applied mathematics, engineering, and related fields, contributing to the advancement of numerical analysis and its applications in solving ODEs.

Multipurpose modified iterative solver for nonlinear equations

2023-07-21
Saher Afshan , Abdul Hanan Sheikh , Fatima Riaz +1 more
Non-linear Eq.s occur as a sub-problem in a wide variety of engineering and scientific domains. To deal with the complexity of Non-linear Eq.s, it is often required to use numerical … Non-linear Eq.s occur as a sub-problem in a wide variety of engineering and scientific domains. To deal with the complexity of Non-linear Eq.s, it is often required to use numerical procedures, which are the most suitable method to employ in certain circumstances. Many classic iterative approaches have been regularly employed for various situations; nevertheless, the convergence rate of those methods is low. In many cases, an iterative approach with a faster convergence rate is needed. This is something that classical methods like the Newton-Raphson Method (NRM) cannot provide. As part of this investigation, a modification to the NRM has been suggested to speed up convergence rates and reduce computational time. Ultimately, this research aims to improve the NRM, resulting in a Modified Iterative Method (MIM). The proposed method was thoroughly examined. According to the research, the convergence rate is higher than that of NRM. The proposed method delivers more accurate results while reducing computational time and requiring fewer iterations than earlier methods. The numerical findings confirm that the promised performance is correct. The results include the number of iterations, residuals, and computing time. This innovative technique, which is appropriate to any Non-linear equation, produces more accurate approximations with less iteration than conventional methods, and it is appropriate to any Non-linear equation.
View PDF Chat

Review of: "New adaptative numerical algorithm for solving partial integro-differential equations"

2024-03-26
Joel N. Ndam
Potential competing interests: No potential competing interests to declare.This work is interesting. Potential competing interests: No potential competing interests to declare.This work is interesting.
View PDF Chat

An Adaptive Semi-Analytical Approach in Solving Nonlinear Korteweg-De Vries Equations

2024-12-31
Abdul Rahman Farhan Sabdin , Che Haziqah Che Hussin , Jumat Sulaiman +2 more
This paper introduces a novel method named the Adaptive Hybrid Reduced Differential Transform Method (AHRDTM) for solving Nonlinear Korteweg-De Vries Equations (NKdVEs). AHRDTM provides convergent semi-analytical solutions over long-time frames … This paper introduces a novel method named the Adaptive Hybrid Reduced Differential Transform Method (AHRDTM) for solving Nonlinear Korteweg-De Vries Equations (NKdVEs). AHRDTM provides convergent semi-analytical solutions over long-time frames by generating subintervals of varying lengths, significantly reducing the number of time-steps and processing time needed for solutions, distinguishing it from the traditional multistep approach of RDTM. Notably, AHRDTM avoids the need for perturbation, linearization or discretization, enhancing its adaptability and reliability. The findings demonstrate that AHRDTM provides highly accurate and efficient solutions for NKdVEs. Additionally, the method is straightforward, significantly reduces the computational effort required to solve NKdVE problems and shows promise for application to a wide range of partial differential equations (PDEs). The efficacy of AHRDTM is illustrated through tables and graphical representations

The Semi Analytics Iterative Method for Solving Newell-Whitehead-Segel Equation

2020-02-24
Busyra Latif , Mat Salim Selamat , Ainnur Nasreen Rosli +2 more
Newell-Whitehead-Segel (NWS) equation is a nonlinear partial differential equation used in modeling various phenomena arising in fluid mechanics.In recent years, various methods have been used to solve the NWS equation … Newell-Whitehead-Segel (NWS) equation is a nonlinear partial differential equation used in modeling various phenomena arising in fluid mechanics.In recent years, various methods have been used to solve the NWS equation such as Adomian Decomposition method (ADM), Homotopy Perturbation method (HPM), New Iterative method (NIM), Laplace Adomian Decomposition method (LADM) and Reduced Differential Transform method (RDTM).In this study, the NWS equation is solved approximately using the Semi Analytical Iterative method (SAIM) to determine the accuracy and effectiveness of this method.Comparisons of the results obtained by SAIM with the exact solution and other existing results obtained by other methods such as ADM, LADM, NIM and RDTM reveal the accuracy and effectiveness of the method.The solution obtained by SAIM is close to the exact solution and the error function is close to zero compared to the other methods mentioned above.The results have been executed using Maple 17.For future use, SAIM is accurate, reliable, and easier in solving the nonlinear problems since this method is simple, straightforward, and derivative free and does not require calculating multiple integrals and demands less computational work.
View PDF Chat

On an iterative Moser-Kurchatov method for solving systems of nonlinear equations

2025-03-26
Ioannis K. Argyros , Stepan Shakhno , Y. V. Shunkin
This paper is devoted to analysis of an iterative method for solving nonlinear equations. The method, inspired by the Kurchatov-type methods, is specifically designed to avoid the need for derivative … This paper is devoted to analysis of an iterative method for solving nonlinear equations. The method, inspired by the Kurchatov-type methods, is specifically designed to avoid the need for derivative calculations or inverses of linear operators. By employing a sequence of approximating operators and divided differences, the method achieves semilocal convergence. Numerical experiments demonstrate the method’s efficiency and robustness, highlighting its potential advantages over traditional methods like Newton’s method, especially in scenarios where derivative calculations are impractical and computationally expensive. The results indicate that the method is a viable and efficient alternative for solving nonlinear equations, especially in large-scale problems or scenarios, where derivative information is not readily available. The robustness and efficiency of the method make it a valuable tool in various scientific and engineering applications.
View PDF Chat

“A Review on the Application of Partial Differential Equations in Various Fields”

2024-01-01
Md. Tuhin Biswas
A fundamental component of contemporary mathematical modeling, partial differential equations (PDEs) is necessary to comprehend the behavior of complex systems in a wide range of scientific fields. This thorough analysis … A fundamental component of contemporary mathematical modeling, partial differential equations (PDEs) is necessary to comprehend the behavior of complex systems in a wide range of scientific fields. This thorough analysis begins with a careful review of the classifications and inherent properties of PDEs before delving into a thorough exploration of their foundations. The work provides a strong basis for future research by classifying PDEs according to their order, linearity, and the makeup of their coefficients. Furthermore, the study explores new directions and difficulties in the field of computational PDEs in addition to explaining current approaches. The paper paves the path for future research advances by addressing important concerns in the field, including stability, convergence, and adaptability. This paper highlights the importance of computational PDEs in developing computational science and engineering by highlighting their crucial function. Through the application of numerical methods, scientists may solve challenging real-world issues with previously unheard-of precision and effectiveness. Furthermore, this work gives practitioners the skills and information needed to push the limits of scientific research and technology innovation by clarifying the complexities of computational PDEs. To sum up, this study shines a light on computational PDEs by offering a thorough road map for negotiating their complex environment and pointing the path toward improved problem-solving abilities, increased scientific understanding, and innovative technology advancements.

Using Proposed Approach to Solve nonlinear Partial Differential Equations

2023-12-10
Noor A. Hussein , Najwan Noori Hani
The purpose of this research is to employ a new method to solve nonlinear differential equations to obtain precise analytical solutions and overcome computation challenges without the need to discretize … The purpose of this research is to employ a new method to solve nonlinear differential equations to obtain precise analytical solutions and overcome computation challenges without the need to discretize the domain or assume the presence of a small parameter, where the method demonstrated a quick and highly accurate solving nonlinear partial differential equations with initial conditions, in compared to existing methods. The phases of the proposed method are straightforward to implement, highly precise, and quickly converge to the correct result.
View PDF Chat

An Overview of Numerical and Analytical Methods for solving Ordinary Differential Equations

2020-01-01
Byakatonda Denis
Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental sciences, social sciences, … Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental sciences, social sciences, banking and many other areas [7]. A differential equation that has only one independent variable is called an Ordinary Differential Equation (ODE), and all derivatives in it are taken with respect to that variable. Most often, the variable is time, t; although, I will use x in this paper as the independent variable. The differential equation where the unknown function depends on two or more variables is referred to as Partial Differential Equations (PDE). Ordinary differential equations can be solved by a variety of methods, analytical and numerical. Although there are many analytic methods for finding the solution of differential equations, there exist quite a number of differential equations that cannot be solved analytically [8]. This means that the solution cannot be expressed as the sum of a finite number of elementary functions (polynomials, exponentials, trigonometric, and hyperbolic functions). For simple differential equations, it is possible to find closed form solutions [9]. But many differential equations arising in applications are so complicated that it is sometimes impractical to have solution formulas; or at least if a solution formula is available, it may involve integrals that can be calculated only by using a numerical quadrature formula. In either case, numerical methods provide a powerful alternative tool for solving the differential equations under the prescribed initial condition or conditions [9]. In this paper, I present the basic and commonly used numerical and analytical methods of solving ordinary differential equations.
View PDF Chat

An Overview of Numerical and Analytical Methods for solving Ordinary Differential Equations

2020-12-10
Byakatonda Denis
Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental sciences, social sciences, … Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental sciences, social sciences, banking and many other areas [7]. A differential equation that has only one independent variable is called an Ordinary Differential Equation (ODE), and all derivatives in it are taken with respect to that variable. Most often, the variable is time, t; although, I will use x in this paper as the independent variable. The differential equation where the unknown function depends on two or more variables is referred to as Partial Differential Equations (PDE). Ordinary differential equations can be solved by a variety of methods, analytical and numerical. Although there are many analytic methods for finding the solution of differential equations, there exist quite a number of differential equations that cannot be solved analytically [8]. This means that the solution cannot be expressed as the sum of a finite number of elementary functions (polynomials, exponentials, trigonometric, and hyperbolic functions). For simple differential equations, it is possible to find closed form solutions [9]. But many differential equations arising in applications are so complicated that it is sometimes impractical to have solution formulas; or at least if a solution formula is available, it may involve integrals that can be calculated only by using a numerical quadrature formula. In either case, numerical methods provide a powerful alternative tool for solving the differential equations under the prescribed initial condition or conditions [9]. In this paper, I present the basic and commonly used numerical and analytical methods of solving ordinary differential equations.

Review of: "New adaptative numerical algorithm for solving partial integro-differential equations"

2023-11-24
Rafid Habib Buti
Potential competing interests: No potential competing interests to declare.The paper is very good ,The paper contains new ideas, as well as the research language is good.The author touched on new … Potential competing interests: No potential competing interests to declare.The paper is very good ,The paper contains new ideas, as well as the research language is good.The author touched on new equations on the subject of differential equations.The researcher also discussed a new idea related
View PDF Chat

A Friendly Iterative Technique for Solving Nonlinear Integro-Differential and Systems of Nonlinear Integro-Differential Equations

2017-07-19
A. A. Hemeda
In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential … In this work, a simple new iterative technique based on the integral operator, the inverse of the differential operator in the problem under consideration, is introduced to solve nonlinear integro-differential and systems of nonlinear integro-differential equations (IDEs). The introduced technique is simpler and shorter in its computational procedures and time than the other methods. In addition, it does not require discretization, linearization or any restrictive assumption of any form in providing analytical or approximate solution to linear and nonlinear equations. Also, this technique does not require calculating Adomian’s polynomials, Lagrange’s multiplier values or equating the terms of equal powers of the impeding parameter which need more computational procedures and time. These advantages make it reliable and its efficiency is demonstrated with numerical examples.

Review of: "New adaptative numerical algorithm for solving partial integro-differential equations"

2023-11-23
Y. H. Youssri
The abstract and introduction clearly outline the paper's objective of introducing a numerical approach based on orthonormal Bernoulli polynomials for solving parabolic partial integro-differential equations (PIDEs).The paper contributes to the … The abstract and introduction clearly outline the paper's objective of introducing a numerical approach based on orthonormal Bernoulli polynomials for solving parabolic partial integro-differential equations (PIDEs).The paper contributes to the field by providing operational matrices and a transformation strategy for solving PIDEs, enhancing the understanding of numerical techniques in this domain.2. The paper establishes a solid theoretical foundation by introducing operational matrices for orthonormal Bernoulli polynomials.This adds mathematical rigor to the proposed numerical approach, contributing to the credibility of the presented method.3. The approach's transformation of the PIDE problem into a nonlinear algebraic system is well-explained, simplifying the computational process and providing clarity in the methodology.4. The inclusion of a convergence analysis is commendable, providing insights into the reliability and accuracy of the proposed numerical algorithm.This analysis enhances the paper's scientific rigor. The paper goes beyond presenting the proposed technique by including a comparison with other well-known methods.This comparative analysis adds value by benchmarking the new approach against existing ones, aiding readers in assessing its effectiveness.6.The conclusion appropriately highlights the versatility of the proposed algorithm, indicating its potential application to various types of PIDEs and differential equations.This broad scope enhances the practical utility of the presented methodology.7. The inclusion of numerical results and test problems, along with comparisons with other algorithms, is a positive aspect.It provides practical evidence of the proposed approach's efficiency and effectiveness.8. The paper emphasizes that the presented method is easily implementable and simple, making it accessible to a broader audience and promoting its practical usability.9.The conclusion encourages the extension of the proposed algorithm to more dimensions, showcasing a forwardthinking approach and suggesting potential avenues for future research.
View PDF Chat

An Enhanced Temimi-Ansari Method for Solving Nonlinear Fredholm Integro-Differential Equations

2025-04-16
M. N. Nyikyaa , A. M. Kwami , A. G. Madaki +1 more
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.

Modified Laplace-variational iteration method for solving linear and nonlinear Volterra integro-differential equations

2025-02-14
Endurance Emagbeweta , J. S. Apanapudor
This study presents a modified Laplace-variational iteration method (MLVIM) designed to solve linear and nonlinear Volterra integro-differential equations (VIDEs) with specified initial conditions. The MLVIM is a hybrid approach that … This study presents a modified Laplace-variational iteration method (MLVIM) designed to solve linear and nonlinear Volterra integro-differential equations (VIDEs) with specified initial conditions. The MLVIM is a hybrid approach that integrates the strengths of the Laplace transform and the variation iteration method (VIM), effectively enhancing the overall solution process by improving both the efficiency and convergence rate. Specifically, the method refines the correction functional and optimizes the handling of the integral term, which directly leads to a reduction in the number of iterations needed and decreases the associated computational complexity. To demonstrate the effectiveness of MLVIM, the study applies it to two illustrative examples involving both linear and nonlinear VIDEs, with initial conditions. The results are then compared to those obtained using the Adomian decomposition method (ADM) and the fourth-order Runge-Kutta (RK4) algorithm. The findings show that MLVIM consistently exhibits a faster convergence rate and higher accuracy compared to both ADM and RK4 in all the examples presented. The MLVIM can be applied to a broad range of linear and nonlinear VIDEs. This makes it a valuable tool with potential applications in various scientific and engineering fields, where integro-differential equations frequently arise in modeling complex systems and processes.
View PDF Chat

An Enhanced Variational Iteration Method for Solving Ordinary and Partial Differential Equations

2025-04-07
A. Hassan , M. Y. Adamu , A. G. Madaki +3 more
The Variational Iteration Method (VIM) has proven to be a powerful technique for solving both ordinary and partial differential equations. However, its reliance on Lagrange multipliers for each type of … The Variational Iteration Method (VIM) has proven to be a powerful technique for solving both ordinary and partial differential equations. However, its reliance on Lagrange multipliers for each type of equation has posed significant limitations, complicating its application and reducing its efficiency. This study introduces a Modified Variational Iteration Method (MVIM) that eliminates the need for Lagrange multipliers, addressing these challenges. The MVIM reformulates the correctional functional, simplifying the solution process and enhancing computational efficiency. The method is applied to both linear and nonlinear ordinary and partial differential equations, demonstrating its ability to provide accurate and fast-converging solutions. Numerical examples show that the MVIM outperforms traditional VIM in terms of computational time and convergence speed, and compares favourably with other methods such as the Adomian Decomposition Method (ADM) and New Iteration Method (NIM). The results highlight the potential of MVIM as a versatile and efficient tool for solving complex differential equations in a variety of scientific and engineering applications.

Numerical methods

2014-03-20
Anders Rasmuson , Bengt Andersson , Louise Olsson +1 more
Differential equations play a dominant role in mathematical modeling. In practical engineering applications, only a very limited number of them can be solved analytically. The purpose of this chapter is … Differential equations play a dominant role in mathematical modeling. In practical engineering applications, only a very limited number of them can be solved analytically. The purpose of this chapter is to give an introduction to the numerical methods needed to solve differential equations, and to explain how solution accuracy can be controlled and how stability can be ensured by selecting the appropriate methods. The mathematical framework needed to solve both ordinary and partial differential equations is presented. A guideline for selecting numerical methods is presented at the end of the chapter.

Innovative Methods for Numerical Solution of Partial Differential Equations

2001-01-01