diffrential eqation basics and application .pptx
- 1. INTRODUCTION • Definition of a differential equation • Importance in various disciplines (engineering, physics, biology, economics) • Two main types: Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) • Applications include population modeling, fluid dynamics, and heat transfer
- 2. TYPES OF DIFFERENTIAL EQUATIONS • Ordinary Differential Equations (ODEs): Involves derivatives with respect to a single variable • Example: dy/dx = 2sin(x) • Common applications: Mechanical vibrations, circuit analysis • Partial Differential Equations (PDEs): Involves derivatives with respect to multiple variables • Example: ∂²u/∂x² + ∂²u/∂y² = 0 • Used in heat conduction, wave equations, fluid mechanics
- 3. ORDER AND DEGREE OF DIFFERENTIAL EQUATIONS • Order: Highest derivative in the equation • Example: d²y/dx² + 5(dy/dx)³ - 4y = e^x (2nd order) • Degree: Power of the highest order derivative (after removing radicals and fractions) • Example: d³y/dx³ + (d²y/dx²)³ + y = 7 (Degree = 1) • Important for classifying equations and choosing appropriate solution techniques
- 4. LINEARVS. NONLINEAR DIFFERENTIAL EQUATIONS • Linear Equations: No products or powers of and its derivatives • Example: dy/dx + xy = e^x • Common in physics and engineering • Nonlinear Equations: Includes products and powers • Example: y(dy/dx) + x = sin(x) • Often require numerical methods for solutions
- 5. HOMOGENEOUSVS. NON-HOMOGENEOUS EQUATIONS • Homogeneous: g(x) = 0, • Example: y' + y = 0 • Solutions often involve exponential functions • Non-Homogeneous: g(x) ≠ 0, • Example: y' + 2y = x • Requires additional methods like undetermined coefficients
- 6. INITIAL AND BOUNDARYVALUE PROBLEMS • InitialValue Problem (IVP) • Differential equation with an initial condition y(x0) = y0 • Example: y' = x, y(0) = 1 • Used in physics for motion analysis • BoundaryValue Problem (BVP) • Conditions given at multiple points • Example: y'' + p(x)y' + q(x)y = r(x), y(a) = 1, y(b) = 2 η η • Applied in structural analysis and thermal conductivity
- 7. WELL-POSED PROBLEMS • A problem is well-posed if: 1. A solution exists 2. The solution is unique 3. The solution is stable (small changes in input lead to small changes in output) • Ensures meaningful physical and engineering applications
- 8. METHODS OF SOLVING DIFFERENTIAL EQUATIONS • Analytical Methods: • Variable separable method • Undetermined coefficients • Variation of parameters • Power series method • Laplace transform method • Numerical Methods: • Euler methods (forward and backward) • Runge-Kutta methods (higher accuracy) • Multi-step methods (Predictor-Corrector, Implicit/Explicit Multistep) • Used for solving complex real-world problems
- 9. NUMERICAL ERRORS • Inherent Error:Arises due to simplifications in modeling • Round-off Error: Due to limited precision in computing devices • Truncation Error: Due to approximating infinite series • Methods to minimize errors:Adaptive step sizing, higher-order methods
- 10. SINGULAR PERTURBATION PROBLEMS • Occurs when a small parameter affects the highest order derivative • Types: • Convection-diffusion: Order reduces by 1 when E = 0 • Reaction-diffusion: Order reduces by 2 when E = 0 • Boundary layers: Rapid changes in solutions in small regions • Important in fluid dynamics and heat transfer
- 11. SOLVING SINGULAR PERTURBATION PROBLEMS • Asymptotic Methods:Approximate solutions using perturbation expansions • Numerical Methods: • Standard finite difference methods • Fitted operator methods • Fitted mesh methods (Shishkin mesh) • Improve accuracy in layer regions
- 12. DELAY DIFFERENTIAL EQUATIONS (DDES) • Derivative depends on past values • Example: y'(t) = f(t, y(t), y(t- )) τ • Applications: • Population dynamics • Disease modeling • Neural networks and signal processing
- 13. SHISHKIN MESH FOR SINGULAR PERTURBATION PROBLEMS • Piecewise uniform mesh designed for handling boundary layers • Dense in layer regions, coarse elsewhere • Separates layer region from smooth region • Improves numerical stability and convergence
- 14. CONCLUSION • Differential equations model real-world systems in science and engineering • Various analytical and numerical techniques exist • Singular perturbation and delay differential equations require special handling • Shishkin mesh and finite difference methods provide effective solutions • Future research includes adaptive meshes and machine learning approaches
- 15. REFERENCES • Cited works related to singular perturbation problems and delay differential equations • Overview of existing numerical methods and recent advancements