Vol 13 Issue 1 April 2025-September 2025

Abstract: Laplace's equation is a pivotal partial differential equation (PDE) in mathematical physics, with applications spanning electrostatics, steady-state heat transfer, fluid mechanics, and gravitational fields. It models systems in equilibrium, where quantities like temperature or electric potential remain constant over time in the absence of sources. In two dimensions, Laplace's equation is written as:

∂2u∂x2+∂2u∂y2=0

where u(x, y)  denotes the potential at coordinates (x, y) . In three dimensions, it includes an additional term for the z -coordinate:

∂2u∂x2+∂2u∂y2+∂2u∂z2=0.

To solve this equation analytically, appropriate boundary conditions must be defined, such as Dirichlet (prescribed values) or Neumann (prescribed derivatives) conditions. Among the most effective methods for tackling Laplace's equation in regular geometries is separation of variables, which transforms the PDE into simpler ordinary differential equations (ODEs). This article delves into the separation of variables technique, focusing on its application to a two-dimensional Laplace's equation in a rectangular domain, with a detailed derivation, key assumptions, and a practical example.

Keywords: Laplace's equation, partial differential equation (PDE), mathematical physics.

Title:  Analytical Solution of Laplace's Equation via Separation of Variables

Author: Ahmad M. D. Al-Eybani

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 50-54

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Website: www.researchpublish.com

Published Date: 08-May-2025

Abstract: Delay differential equations (DDEs) are a class of differential equations where the derivative of the unknown function at a given time depends not only on the current state but also on its values at previous times. These equations arise in numerous fields, including biology, engineering, and economics. Unlike ordinary differential equations (ODEs), DDEs incorporate time delays, making their solution more complex due to the need for a history function. A common form of a first-order linear DDE with a constant delay is:

dy(t)dt=-ay(t)+by(t-τ),

where y(t)  is the unknown function, a  and b  are constants, and τ>0  is the time delay. Solving DDEs analytically or numerically requires specialized techniques, and one of the most straightforward and intuitive methods for constant-delay DDEs is the method of steps. This approach breaks the problem into sequential intervals, solving an ODE in each interval using the solution from the previous interval as a history function. In this article, we explore the method of steps in detail, including its mathematical derivation, assumptions, practical implementation, and an illustrative example.

Keywords: Delay differential equations (DDEs), numerous fields, including biology, engineering, economics.

Title: Solving Delay Differential Equations Using the Method of Steps

Author: Ahmad M. D. Al-Eybani

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 46-49

Research Publish Journals

Website: www.researchpublish.com

Published Date: 08-May-2025

Abstract: The integration of supplemental videos in teaching mathematics allows students to comprehend more and recall key purposes of the discussion. Those enhancements speak to some significant learning results. This study wanted to investigate the effectiveness of integrating supplemental videos in teaching mathematics. The video clips presented to the respondents during class were harvested from youtube.com. A researcher-made pretest posttest was used as the main instrument in this study. An adopted survey on the level of self- efficacy was also given to the respondents of the experimental and control group to determine the capacity of the students to solve mathematics problems. Results indicate that the performance level of the students from the experimental group has increased significantly after the exposure of the supplemental videos. However, there is no significant difference on the performances between the control and experimental group on their posttest results, hence, it can be concluded that the performances of both groups are comparable. Survey results on the level of self – efficacy of the experimental group also shows that there was no change on the beliefs of the students on their capacity in terms of solving problems in mathematics. The given results have been confirmed with a subjective assessment of students’ regard for learning with the use of supplemental videos. Further study that led to discover the impact on the use of supplemental videos to another group of students is recommended

Keywords: mathematics, performance, self- efficacy, students’ regard, supplemental videos.

Title: SUPPLEMENTAL VIDEOS IN TEACHING MATHEMATICS

Author: Margie M. Oacan

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 35-45

Research Publish Journals

Website: www.researchpublish.com

Published Date: 08-May-2025

Abstract: The heat equation is one of the fundamental partial differential equations in mathematical physics, describing how heat diffuses through a medium over time. It appears in various fields, such as thermodynamics, fluid dynamics, and materials science, and is a cornerstone of understanding diffusive processes. Mathematically, the heat equation in one spatial dimension is expressed as:

∂u∂t=α∂2u∂x   ,

where (u(x, t)  represents the temperature at position (x)  and time (t) , and (α)  is the thermal diffusivity, a positive constant that depends on the material properties. Solving this equation analytically can be challenging due to its dependence on both space and time, but one of the most elegant and widely applicable methods is separation of variables. This technique reduces the PDE into a set of ordinary differential equations (ODEs), which are easier to solve. In this article, we will explore the separation of variables method in detail, including its assumptions, step-by-step derivation, and a practical example applied to a finite rod with fixed boundary conditions.

Keywords: heat equation, thermodynamics, fluid dynamics, variables method.

Title: Solving the Heat Equation Using Separation of Variables

Author: Ahmad M. D. Al-Eybani

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 31-34

Research Publish Journals

Website: www.researchpublish.com

Published Date: 29-April-2025

Abstract: The Burgers equation is a fundamental partial differential equation (PDE) in applied mathematics, mathematical physics, and engineering, describing a variety of nonlinear wave phenomena such as fluid dynamics, gas dynamics, and traffic flow. Its nonlinear nature makes it a challenging yet fascinating problem to solve analytically or numerically. Among the various methods developed to tackle the Burgers equation, the Differential Transform Method (DTM) has emerged as a powerful semi-analytical technique due to its simplicity, computational efficiency, and ability to handle nonlinear PDEs. This article provides an in-depth exploration of the DTM and its application to solving the Burgers equation, covering its theoretical foundations, implementation, advantages, limitations, and illustrative examples.

Keywords: partial differential equation (PDE), Differential Transform Method (DTM),  Burgers equation.

Title: The Differential Transform Method for Solving the Burgers Equation

Author: Ahmad M. D. Al-Eybani

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 26-30

Research Publish Journals

Website: www.researchpublish.com

Published Date: 29-April-2025

Abstract: The Burgers equation, a fundamental partial differential equation (PDE) in applied mathematics and physics, is widely studied due to its applications in fluid dynamics, gas dynamics, traffic flow, and nonlinear wave propagation. Named after Johannes Martinus Burgers, this equation combines nonlinear convection and diffusion, making it a simplified model for complex phenomena like turbulence and shock waves. The one-dimensional Burgers equation is typically expressed as:

∂u∂t+u∂u∂x= ν∂2u∂x2

where  u(x,t)  represents the velocity field, ν>0   is the kinematic viscosity, t  is time, and x  is the spatial coordinate. The nonlinear term u∂u∂x   introduces complexity, making analytical solutions challenging except in specific cases. As a result, semi-analytical and numerical methods have been developed to approximate solutions to the Burgers equation.

Among these methods, the Differential Transform Method (DTM) and the Adomian Decomposition Method (ADM) are two powerful semi-analytical techniques that have gained attention for their ability to handle nonlinear PDEs efficiently. This article provides a detailed comparison of DTM and ADM when applied to the Burgers equation, examining their theoretical foundations, implementation procedures, advantages, limitations, and performance in terms of accuracy, computational efficiency, and applicability.

Keywords: Differential Transform Method (DTM), Adomian Decomposition Method (ADM), Burgers equation.

Title: A Comparative Analysis of the Differential Transform Method and the Adomian Decomposition Method for Solving the Burgers Equation

Author: Ahmad M. D. Al-Eybani

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 20-25

Research Publish Journals

Website: www.researchpublish.com

Published Date: 29-April-2025

Abstract: The presence of total coliform bacteria in rivers represents a serious hazard to health and the environment, causing a highlight on the need for sustainable treatment options regarding water. This study examines bamboo (Bambusa vulgaris “Wamin”) fiber strands in bamboo and their role in reducing total coliform from river water as a natural filtration media. Due to sustainable, fast-growing and naturally antibacterial features of bamboo, it makes a resourceful material for treatment solutions on water without harming the environment. In this study, the developed filtration media is a process combining bamboo fiber strands grinding and carbonization which is aimed at biofiltration maximization. These observations support the idea that, baring environmental concerns, treated bamboo material can be an effective and very cheap strategy for water purification purposes by carbonizing bamboo in a cost-effective manner to remove disease-causing organisms from water. Such chemical limitations underscore the need for additional research to focus on particular effective dosages and the shelf-life of the filtration material made of bamboo for practical faster performance.

Keywords: Bamboo Fiber, Carbonized Bamboo, Total Coliform, Natural Filtration, River Water.

Title: Natural Filtration: Exploring the Efficacy of Bamboo (Bambusa Vulgaris “Wamin”) Fiber Strand with Carbonized Bamboo for the Reduction of Total Coliform in River Water

Author: CARBONEL, MARK ANDREW A., ALVAR, EMANUEL C., MARQUEZ, JEROME IVAN U., PAUL CHRISTIAN YBARRA PALBAN, DR. SHERWIN FORTUGALIZA

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 8-19

Research Publish Journals

Website: www.researchpublish.com

Published Date: 14-April-2025

Abstract: The heat equation is a fundamental partial differential equation (PDE) in mathematical physics, widely used to model the diffusion of heat in each medium over time. It arises in diverse fields such as engineering, physics, and even financial mathematics (where it models option pricing under the Black-Scholes framework). Solving the heat equation efficiently and accurately is a central problem in applied mathematics, and numerous methods have been developed to tackle it. Among these, two prominent approaches stand out: the Finite Difference Method (FDM) and the Finite Element Method (FEM). This article provides an in-depth comparison of these two methods, exploring their theoretical foundations, computational implementations, advantages, limitations, and practical applications. By the end, readers will have a clear understanding of how these methods differ and when each might be preferable.

Keywords: mathematical physics, financial mathematics, Finite Element Method (FEM).

Title: A Comprehensive Comparison of Two Methods for Solving the Heat Equation

Author: Ahmad M. D. Al-Eybani

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 3-7

Research Publish Journals

Website: www.researchpublish.com

Published Date: 09-April-2025

Abstract: To describe nature, we must set principles! Nature working with the principle of the infinite fraction of matter, only it has just been detected by the accelerators!

Until now, the material has operated on the principle of the minimum indivisible. And the least indivisible thing is the bubbles-particles, which I described in my atomic physics and worldview! My worldview is the book prophesied by John in REVELATION!

Subatomic particles can have a charge or not. For example, neutrinos don't have!

There are well-founded suspicions that nature is heading towards the beginning of the infinite part!

Keywords: Nature working, infinite fraction, atomic physics.

Title: THE NEW NATURE WITH THE BEGINNING OF THE INFINITE PART OF MATTER

Author: ALEKOS CHARALAMPOPOULOS

International Journal of Mathematics and Physical Sciences Research  

ISSN 2348-5736 (Online)

Vol. 13, Issue 1, April 2025 - September 2025

Page No: 1-2

Research Publish Journals

Website: www.researchpublish.com

Published Date: 01-April-2025