Fitted computational method for solving singularly perturbed small time lag problem.
Sisay Ketema Tesfaye1, Mesfin Mekuria Woldaregay2, Tekle Gemmechu Dinka1
1Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia.
BMC Research Notes
|October 11, 2022
Summary
A new exponentially fitted numerical method accurately solves singularly perturbed time lag problems with boundary layers. This method offers improved accuracy over existing schemes for these complex differential equations.
Area of Science:
- Numerical Analysis
- Computational Mathematics
- Differential Equations
Background:
- Singularly perturbed problems with time lags present unique challenges due to boundary layers.
- Accurate numerical solutions are crucial for understanding phenomena governed by these equations.
Purpose of the Study:
- To develop an accurate exponentially fitted numerical method for singularly perturbed time lag problems.
- To analyze the stability and convergence properties of the proposed method.
- To demonstrate the method's superior accuracy compared to existing techniques.
Main Methods:
- Applied the backward-Euler method for temporal discretization.
- Employed a higher-order finite difference method for spatial discretization.
- Incorporated an exponential fitting factor into the difference scheme for stability.
Main Results:
- The method exhibits uniform convergence with a linear order.
- Stability analysis using the comparison principle confirms the method's robustness.
- Numerical examples demonstrate superior accuracy compared to literature results.
Conclusions:
- The proposed exponentially fitted method is accurate and stable for solving singularly perturbed time lag problems.
- This approach effectively handles the boundary layer behavior inherent in these problems.
- The method provides a reliable and more accurate alternative for computational analysis.
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