Partial differential equations involve results of unknown functions when there are multiple independent variables. There is a need for analytical solutions to ensure partial differential equations could be solved accurately. Thus, these partial differential equations could be solved using the right initial and boundaries conditions. In this light, boundary conditions depend on the general solution; the partial differential equations should present particular solutions when paired with varied boundary conditions. This study analysed the use of variable separation to provide an analytical solution of the homogeneous, one-dimensional heat equation. This study is applied to varied boundary conditions to examine the flow attributes of the heat equation. The solution is verified through different boundary conditions: Dirichlet, Neumann, and mixed-insulated boundary conditions. the initial value was kept constant despite the varied boundary conditions. There are two significant findings in this study. First, the temperature profile changes are influenced by the boundary conditions, and that the boundary conditions are dependent on the heat equation’s flow attributes.

Heat equation is a partial differential equation that contains derivatives for two or more independent variables of an unknown function. There are heat equations that are homogenous and non-homogeneous. The non-homogeneous heat equation can be defined as there are source term in the partial differential equations, while there is no source term for the homogeneous heat equation. To solve the nonhomogeneous, the heat equation need to be homogeneous by measure the displacement of the temperature of the heat from the equilibrium temperature. To obtain the exact solution of partial differential equation, an analytical solution is required. The suitable boundary and initial conditions are required to solve these partial differential equations. In the present research, the homogeneous and nonhomogeneous one-dimensional heat equation will be solved analytically by using superposition principle (non-homogeneous) and separation of variables method (homogeneous). Our main objective is to compare the flow characteristics of heat equation on homogeneous and non-homogeneous heat equation. The heat equation will be solved based on Dirichlet boundary conditions to verify our objective. The finding results have been compared on the temperature profile for homogeneous and non-homogeneous heat equations. The temperature profile on the non-homogeneous Dirichlet boundary conditions show that it can be changed the profile and remain the same as temperature profile on homogeneous Dirichlet boundary condition when there is no source term.

Designing the Kelarai pattern is not easy and it is requires mathematical skills to ensure a consistent design. Three key skills are required to produce the Kelarai design which are creatively to shape the pattern, ability to calculate the weave angle and arrange the colours, and efficiency in weaving to ensure that weave overlays are done consistently. In Mathematics, rotation, reflection, and enlargement are types of transformation concept to make some changes in any given geometric shape. In this research, the concept of transformation on pandan mat weaving art at Kolej GENIUS Insan was analysed. The main objective is to determine the pattern of Kelarai Kepala Gajah between the combined transformation Z, followed by Y, X and W that also can be written as transformation WXYZ. Under the combined transformation, Z is defined as an anticlockwise rotation of 90° about the center origin. After rotation, the image formed was reflected using Y and X transformation, which are reflection on line x = 0 and on line y = x, respectively. The transformation W described the final image of Kelarai Kepala Gajah pattern, where the concept of enlargement with scale factor 2 at center of (7, -7) was applied. However, the enlargement with scale factor 2 at center (7, 7) in a single transformation shows the same results as the final image under combined transformation WXYZ. The uniqueness of Kelarai Kepala Gajah pattern can be shown when the final image remains the same for both combined and single transformation were applied.

Science and Mathematics are very enjoyable subjects if students can learn, understand, and apply them daily. However, many students are not interested in Mathematics compared to other subjects because it requires logical thinking and proving of basic concepts, where most students find it difficult to understand. A few gifted and talented Muslim students have a poor level of achievement in Mathematics. They think that Mathematics is a boring subject and thus unimportant in daily life. The main objective of this research is to develop Al-Khawarizmi Number Base Module for gifted and talented Muslim students at Kolej PERMATA Insan, Universiti Sains Islam Malaysia. This module helps students to enhance their understanding and achievement in Mathematics. This module is divided into four themes, namely Number Basis, Complex Number, Linear Equations and Algebra. The students involved in this module are assessed through four assessment methods which are, first, students must complete all the module given based on specific topics. Second, students must play a game card based on specific topics for analyzing their speed-thinking and understanding level. Then, students are also assessed through a pre and post-test for the topics that they have learned in the module. Lastly, students should fill out a survey form to find out their level of improvement and achievement in Mathematics. Improving the understanding of knowledge of mathematics among gifted and talented Muslim students will also encourage students to cultivate the knowledge of mathematics throughout their lives as well.

This work focuses on the development of a mathematical model as a viable alternative to pinpoint locations of gas leaks in a pipeline. The transient non-isothermal flow of hydrogen-natural gas mixture is considered because hydrogen is often transported in the same pipeline as natural gas to reduce the transportation cost. The mathematical model developed took into consideration the effect of the mass ratio of gas mixture. The gas mixture was assumed to be homogeneous and the transient pressure wave was created by the sudden or instantaneous closure of a downstream shut-off valve to ensure the attainment of minimum pressure at the downstream end within a short time. The governing equations were numerically solved using the reduced order modelling (ROM) technique, which had not been previously applied on non-isothermal models involving gas mixtures. Numerical results observed that the mass ratio of hydrogen to natural gas should not be more than 0.5 to ensure that leakage does not occur before the estimated leak position. An increase in the mass ratio leads to an increase in the pressure and celerity wave, while the leak location and the amount of leak discharge decrease.

Science and Mathematics are very enjoyable subjects if students can learn, understand and apply them daily. However, many students are less interested in this Mathematics subject than others such as language, history and physical and health education. This is because Mathematics requires students to think logically through proofs and various basic concepts, which most students find difficult to understand. A few gifted and talented Muslim students have a poor level of achievement in Mathematics. They think that Mathematics is boring and not important in daily life. A special module named the Al-Khawarizmi Number Base Module is used as an intervention step for students who need guidance and support to achieve excellence in Mathematics. The main objective in this research is to verify the effectiveness of the Al-Khawarizmi Basic Number Module on the achievement and understanding of Mathematics among gifted and talented Muslim students at PERMATA Insan College, Universiti Sains Islam Malaysia. This module is divided into three categories: 28% of students are required to follow the entire module as an intervention process, 30% of students can follow some parts of the module, and 42% of students are capable of being facilitators during intervention workshops. Achievement of students' understanding in Mathematics can produce Muslim scholars who are future-ready and excel in the field of Mathematical Sciences in line with the intentions of the PERMATA Insan College Strategic Plan and Malaysian Education Development Plan 2013-2025.

GeoGebra is one of the teaching tool conducted by educator implement into their lesson plans to improve teaching and learning quality, make the classroom lesson more enjoyable, creative and innovative. Students can construct a graph, modify and observe the effects of changing graph shape. Moreover, students can be saved all their works documents for future reference systematically. The aim of this research was to analyse the application of GeoGebra on mathematics achievement in relation to the quadratic functions among gifted and talented Muslims student’s at Kolej GENIUS Insan, Universiti Sains Islam Malaysia. The effects of changing the coefficients values on the graphs of quadratic functions was identified. The results illustrate that changing in value of coefficient aon quadratic functions affects the shape and width of the graph, but, not the y-intercept. The position of the vertex on y-axis is only affected by changing the value of coefficient b, but the shape of the graph and the y-intercept remain unchanged. The shape of the graph is unaffected by changing the value of coefficient c, which only affects the location of the graph vertically upwards or downwards. Students' performance in understanding the effect of quadratic graphs improves when they use GeoGebra software.

GeoGebra is a teaching tool that educators use in their lesson plans to improve the quality of teaching and learning. Instead of drawing on a sheet of paper, students can design a graph, adjust the actual graph shape, and examine the impact of changing graph pattern using GeoGebra in mathematics teaching and learning. Furthermore, students can keep all of their work materials in a structured manner for future reference. GeoGebra will make a school lecture more interesting, exciting, creative, and innovative. The goal of this study was to analyse the effects of GeoGebra software in Mathematics achievement in respect to quadratic functions among gifted and talented Muslims student’s at Kolej GENIUS Insan, Universiti Sains Islam Malaysia. The maximum or minimum point of quadratic function was determined by using GeoGebra software, and the characteristics of quadratic expressions in one variables was also identified. The results illustrate that the graph quadratic expression has the highest point or the lowest point based the values of coefficient a on the quadratic function. For the graph function with negative values of coefficient a on a quadratic function, there are highest values of coordinates x and y, which also known as maximum point, while the graph function with positive values of a on a quadratic function, there are lowest values of coordinates x and y, which also known as minimum point. When students utilize GeoGebra software, their performance in calculating the minimum and maximum points on quadratic functions improves.

There are a few bright and intelligent students who have a poor level of achievement in mathematics. They think that mathematics is a boring subject and not important in daily life. A special module named the Al-Khawarizmi Number Basis Module will be used as an intervention step for students who need this guidance and support to achieve excellence in mathematics, increase their level of understanding of professional mathematics, and give encouragement to Muslim students in cultivating a mathematical life. The main objective of this research is to develop a game card for gifted and talented Muslim students at Kolej PERMATA Insan, Universiti Sains Islam Malaysia. This module helps students enhance their understanding and achievement in mathematics. This module contains three main topics, namely rational numbers, factors and multiples, and standard form. The effectiveness of this module needs to be evaluated and analyzed carefully. This way, the module can improve the level of achievement and understanding of gifted and talented Muslim students, which can then be shared with students that are in mainstream schools. In completing this module, students are also tested with a game card called Find and Match: Al-Khawarizmi Robot of Number (Riz and Ron). Then, students will undergo a pre-test and post-test to assess their level of achievement. The implementation of this module can increase understanding and strengthen the professional knowledge of mathematics among gifted and talented Muslim students and encourage students to cultivate mathematical knowledge throughout life.

Hand is the main organ used to manipulate the environment and to perform daily activities. Every human is endowed with a pair of right and left hands. A person's tendency to use the left or right hand can usually be clearly seen during childhood, which is from the age of two. Most children perform various activities and hold objects automatically with the right hand. However, there are a few individuals who are born with a tendency to use left-handedness. They feel that the left hand is stronger than the right hand and they are more comfortable to do many things. This group of individuals is called a recessive gene. The choice of hand use is based on a person's genetic variation. Previous studies have proven several factors that influence the preference of hand use such as environment, gender, and genes. The main purpose of this study is to analyse the genetic factors that influence the use of left-handedness on gifted and talented Muslim students at PERMATA Insan College. Based on the survey, the results show that majority of left-handed students have a family background of left-handedness and inherited from their parents. The advantage of left-handed individuals is that they have a high ability in the field of mathematics. Achievement of students understanding in Mathematics can produce Muslim scholars who are future-ready and excel in the field of Mathematical Sciences and increase their critical thinking ability.

Transient pressure wave detection analysis to detect the location of leakage on a pipeline containing hydrogen-natural gas mixture is presented. The transient pressure wave is generated either by rapid or sudden closure of the downstream shut-off valve. The governing equations of unsteady, compressible and isothermal one-dimensional flow are solved using the reduced order modelling technique. The solutions obtained when the transient condition is generated using the rapid closure valve show good agreement with published results. When the sudden closure valve is considered, the transient pressure, celerity wave, mass flux and the amount of leak discharge are shown to increase when the hydrogen mass ratio is increased. The amount of leak discharge which is calculated based on the computed celerity and pressure waves is found to be dependent on the leak positions.

Murid pintar dan berbakat mempunyai kebolehan dalam mempertingkatkan potensi diri untuk memastikan prestasi yang cemerlang dan seterusnya memperoleh pencapaian tertinggi jika dibanding dengan murid lain seusia mereka. Sehubungan itu, murid pintar dan berbakat mendapati sistem pendidikan semasa berada pada tahap yang rendah atau sederhana selaras bedasarkan umur akal mereka. Satu kajian telah dilaksanakan tentang pemetaan kecenderungan sains dan hafazan Al-quran dalam kalangan murid pintar dan berbakat muslim di Malaysia. Soal kaji selidik telah digunakan sebagai penanda aras kecenderungan murid. Seramai 466 orang murid Kolej GENIUS Insan telah dikenalpasti sebagai responden dalam kajian ini. Tujuan kajian ini dijalankan adalah untuk mengenal pasti dan mengelaskan kecenderungan murid terhadap bidang sains yang memfokuskan pemikiran logik dan juga hafalan seperti Matematik, Matematik Tambahan, Fizik, Kimia dan Biologi melalui soal kaji selidik yang telah dijalankan di kalangan murid pintar dan berbakat muslim.

Transient pressure wave detection analysis to detect the location of leakage of non-isothermal flow in an inclined pipeline containing hydrogen-natural gas mixture is investigated. The governing equations are solved using the reduced order modelling technique. The effects of inclination angles, mass ratio of gas mixture and temperature change on the pressure and celerity waves in an inclined pipeline are discussed. The solutions for isothermal flow in a horizontal pipeline show good agreement with published results. For non-isothermal flow an increase in the mass ratio lead to an increase in the pressure and celerity waves, while the leak location and amount of leak discharge decrease. However, it is noted that the amount of leak discharge is still higher than that of isothermal flow. It is also observed that an increase in the inclination angle increases the pressure drop and leak discharge but the celerity wave and the leak location do not seem to be affected. Thus, to reduce the leak discharge, the inclination angle of the pipeline should be reduced and further, to ensure that leakage does not occur before the calculated leak position, the mass ratio of hydrogen to natural gas should not be more than 0.5.

A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The numerical solution is needed to obtain the solution of partial differential equation. To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. In the present study, the homogeneous one-dimensional heat equation will be solved numerically by using Implicit Crank Nicolson method. Our main objective is to determine the flow characteristics of heat equation with Dirichlet boundary condition on homogeneous heat equation. The method of Implicit Crank Nicolson has been chosen because of the stability of the method. The results have been compared with the exact analytical solution. The validated results show that the numerical results remains same as the exact analytical solutions. The results show that the changes of the temperature profile depends on the types of boundary conditions. The boundary conditions will be affected the flow characteristics of the heat equation.