-
Research Article
Solutions of One Dimensional Parabolic Partial Differential Equations: An Improved Finite Difference Approach
Omowo Babajide Johnson*
,
Adeniran Adebayo Oludare,
Adetunkasi Taiwo Flora,
Ogunbanwo Samson Tolulope,
Olatunji Olakunle Henry
Issue:
Volume 13, Issue 4, August 2025
Pages:
237-244
Received:
7 May 2025
Accepted:
6 June 2025
Published:
14 July 2025
DOI:
10.11648/j.ajam.20251304.11
Downloads:
Views:
Abstract: This paper introduces a finite difference scheme derived from the classical Crank-Nicolson method. The proposed scheme offer an improved spatial accuracy while maintaining the second-order temporal accuracy of the original Crank-Nicolson scheme. The higher order of spatial accuracy leads to improved convergence properties. The consistency and stability of the new scheme are analyzed using Taylor series expansion and von Neumann stability analysis, respectively. To validate the efficiency of the proposed scheme, it is implemented in MATLAB to solve the one-dimensional heat equation. To explore the versatility of the scheme, it is further extended to solve the advection-diffusion equation. Numerical experiments demonstrated on diffusion equation show that the new scheme compares favorably with existing methods in terms of convergence and accuracy. The results of the numerical solutions are presented in tabular form to highlight the accuracy and rates of convergence of the method. In addition, graphical plots of the numerical solutions are provided at different time levels to visualize the behavior of the solution over time and to illustrate the consistency between the numerical and analytical results. These visual and numerical comparisons further emphasize the reliability and precision of the proposed scheme. The combination of improved spatial resolution, solid theoretical foundation, and practical implementation demonstrates the schemeˆ as potential for solving time-dependent partial differential equations efficiently and accurately. This makes the scheme a valuable contribution to the field of numerical methods for parabolic-type equations.
Abstract: This paper introduces a finite difference scheme derived from the classical Crank-Nicolson method. The proposed scheme offer an improved spatial accuracy while maintaining the second-order temporal accuracy of the original Crank-Nicolson scheme. The higher order of spatial accuracy leads to improved convergence properties. The consistency and stabi...
Show More
-
Research Article
Homotopy Perturbation Technique for Solving Higher Dimensional Time Fractional Burgers-Huxley Equations
Umesh Kumari*,
Inderdeep Singh
Issue:
Volume 13, Issue 4, August 2025
Pages:
245-255
Received:
11 June 2025
Accepted:
14 July 2025
Published:
31 July 2025
Abstract: Many real-world phenomena such as nerve pulse transmission, fluid transport, and chemical kinetics are modeled using nonlinear partial differential equations with time-fractional derivatives. Among them, the time-fractional Burgers-Huxley equation plays a significant role due to its ability to capture both diffusion and reaction mechanisms with memory effects. Solving such equations in higher dimensions is highly challenging and calls for efficient analytical approaches. In this work we present a technique for handling the time-fractional Burgers-Huxley equation up to k-dimensions by employing Laplace transform based homotopy perturbation method (LT-HPM). The LT-HPM is adopted based on its advantage in dealing with nonlinear terms and simplify the solution process by converting fractional derivatives into more manageable expressions. Unlike other hybrid approaches, LT-HPM is less computational complex and provides rapid convergent series solutions without requiring any linearization or restrictive assumptions. To showcase the effectiveness of this approach, we solve a pair of examples: a 2-D case and a 3-D case. The obtained results confirm that LT-HPM is accurate and powerful in tackling complex nonlinear PDEs in higher dimensions.
Abstract: Many real-world phenomena such as nerve pulse transmission, fluid transport, and chemical kinetics are modeled using nonlinear partial differential equations with time-fractional derivatives. Among them, the time-fractional Burgers-Huxley equation plays a significant role due to its ability to capture both diffusion and reaction mechanisms with mem...
Show More
-
Research Article
Measuring Inequality of Income Distributions in Egypt: An Empirical Study Using Weighted and Non-parametric Methods
Eman Mohamed Hanafy*,
Hend Abdulghaffar Auda,
Ibrahim Hassan Ibrahim
Issue:
Volume 13, Issue 4, August 2025
Pages:
256-273
Received:
23 June 2025
Accepted:
9 July 2025
Published:
4 August 2025
Abstract: One of the foundational tasks in statistical analysis is the design and implementation of sample surveys, which involve sampling error that affects the reliability and precision of the resulting estimates. Measures such as totals and means are insufficient on their own without corresponding indicators of statistical precision, such as confidence intervals. As such, any analytical method applied to survey data must accommodate data weighting, which is essential for producing valid and interpretable estimates. Within the field of economic research, income inequality represents a key application where the use of weighted data is critical. In this context, we introduce a weighted inequality index designed to improve the robustness of inequality measurement. To enhance its analytical rigor, the proposed index is accompanied by a non-parametric, bootstrap-based algorithm, designed to facilitate comparative assessments and statistical significance testing across various population subgroups (e.g., regions, countries, gender). A major advantage of this approach lies in its flexibility; it is suitable for both normally and non-normally distributed data, thereby broadening its applicability to real-world datasets that often deviate from standard distributional assumptions. To demonstrate the empirical utility and comparative performance of the proposed methodology, we applied it to household income data obtained from the Central Agency for Public Mobilization and Statistics (CAPMAS), based on nationally representative income and expenditure surveys conducted in 2015 and 2018. The empirical findings revealed a general decline in the values of the proposed inequality index across most Egyptian governorates between 2015 and 2018, indicating a modest trend toward greater income equality. This downward shift may be indicative of the effects of socioeconomic reform measures and targeted development policies aimed at reducing regional disparities. The results validate the practical relevance of the proposed index as a reliable tool for evaluating income inequality in diverse socioeconomic contexts.
Abstract: One of the foundational tasks in statistical analysis is the design and implementation of sample surveys, which involve sampling error that affects the reliability and precision of the resulting estimates. Measures such as totals and means are insufficient on their own without corresponding indicators of statistical precision, such as confidence in...
Show More
-
Research Article
Modified Spectral Quasilinearization Method for Solving Fluid Flow over Stretching Sheet
Mwatela James Mwakio*,
Samuel Mutua,
Felicien Habiyaremye
Issue:
Volume 13, Issue 4, August 2025
Pages:
274-281
Received:
11 June 2025
Accepted:
30 June 2025
Published:
5 August 2025
Abstract: This study aims to develop an efficient and accurate numerical method for solving the boundary layer fluid flow over a stretching sheet using a modified spectral quasilinearization method. The governing partial differential equations (PDEs) for momentum and energy are first transformed into a system of nonlinear ordinary differential equations (ODEs) using similarity transformations. The improvement in the method of solution is realized by numerically solving the flow equations defined over a larger semi-infinite domain [0, ∞) using spectral quasilinearization method embedded on overlapping sub-intervals. This method is better than its counterpart on a single domain as it maintains high accuracy and at the same time results in a sparse differentiation matrix that is easily invertible and saves CPU time. The numerical simulations and solution error analysis were performed using MATLAB version 2018a. Convergence analysis demonstrates exponential error decay, with residual errors reducing from the order of 10−2 to approximately 10−12 within four iterations, confirming the accuracy and efficiency of the numerical scheme. Additionally, the impact of the Prandtl number on thermal boundary layer thickness is examined, revealing sharper temperature gradients for higher Pr values. This method can be adapted to solve other fluid flow problems represented as systems of nonlinear ODEs.
Abstract: This study aims to develop an efficient and accurate numerical method for solving the boundary layer fluid flow over a stretching sheet using a modified spectral quasilinearization method. The governing partial differential equations (PDEs) for momentum and energy are first transformed into a system of nonlinear ordinary differential equations (ODE...
Show More
