Exponentially Fitted Difference Method For Solving Second Order Singularly Perturbed Volterra Integro-Differential Equations

Abstract

In This Thesis, We Develop And Analyze A Uniformly Convergent Numerical Scheme For Solving Second-Order Singularly Perturbed Linear Volterra Integro-Differential Equations, Which Are Characterized By The Presence Of A Small Perturbation Parameter ?? That Leads To The Formation Of Boundary Layers In The Solution. These Boundary Layers Cause Severe Challenges For Standard Numerical Methods, As They Result In Sharp Gradients Near The Boundaries That Are Difficult To Resolve Accurately Without Excessive Computational Effort. To Address These Challenges, We Propose A Robust Parameter-Uniform Numerical Method That Effectively Captures The Behavior Of The Solution Across Different Scales. The Differential Part Of The Equation Is Discretized Using An Exponentially Fitted Difference Scheme, Which Is Specifically Designed To Handle The Exponential Nature Of The Boundary Layer Solutions By Incorporating The Perturbation Parameter Into The Finite Difference Operator. This Ensures That The Numerical Method Remains Stable And Accurate Even On Coarse Meshes. Meanwhile, The Integral Part Of The Equation Is Approximated Using The Composite Trapezoidal Rule, Which Provides Second-Order Accuracy For Smooth Integrands And Maintains Consistency With The Overall Discretization Strategy. A Rigorous Stability And Convergence Analysis Is Conducted To Establish That The Proposed Scheme Is Uniformly Convergent With Respect To The Perturbation Parameter ??, Meaning That The Accuracy Of The Numerical Solution Does Not Deteriorate As ?? Approaches Zero. To Validate The Theoretical Results And Demonstrate The Effectiveness Of The Method, Two Test Examples With Known Exact Or Highly Accurate Approximate Solutions Are Considered. The Maximum Absolute Errors And Rates Of Convergence Are Computed For Various Values Of ?? And Different Mesh Sizes, And The Results Are Presented In Tabular Form. These Numerical Experiments Confirm That The Proposed Scheme Achieves ??-Uniform Convergence And Efficiently Resolves The Sharp Boundary Layers, Thereby Illustrating Its Reliability And Superiority Over Conventional Methods That Fail To Maintain Accuracy In The Presence Of Strong Boundary Layer Effects.