Engineering Mathematics: Recurrence Relations

Introduction

• Presenter: राजेंद्र पुरोहित
• Focus: Recurrence Relations and Difference Equations
• Target Audience: Engineering Mathematics, B.Sc. students

Key Concepts

Recurrence Relation

• Defined as a relationship where a term is written in relation to its previous terms.
• Example:
  • Recursive sequence: a1, a2, a3, ..., a(n-1), an
  • Recurrence relation: an = 2an-1

Fibonacci Sequence

• Each term is the sum of its previous two terms.
• Example: 1, 1, 2, 3, 5, 8, ...
• Recurrence relation: an = a(n-1) + a(n-2)

Difference Equation

• An equation derived from a recurrence relation.
• Similarity to differential equations but requires different solving techniques.

Order of Recurrence Relation

• Determined by the highest term index difference.
• Example: In an = an-1 + an-2, the order is 2.

Linear Recurrence Relation

• Linearity condition: No term should be a product of other terms (e.g., squared terms).
• Differentiation between linear and non-linear by terms.

Constant Coefficient Linear Recurrence Relations

• Defined by constant coefficients in the relation.
• Solving method involves converting to an auxiliary equation.

Solving Techniques

Auxiliary Equation Method

• Convert recurrence relation into an auxiliary equation.
• Find roots: Real, distinct, and repeated roots determine the solution form.
• Example:
  • Equation: a(n+3) + 2a(n-1) = 0
  • Auxiliary equation: x² + 3x + 2 = 0

Solution Forms

• Real, Distinct Roots: Write general solution as combination of terms raised to power n.
• Repeated Roots: Involve multiplying by n to account for repetition.
• Complex Roots: Use polar form and trigonometric identities.

Coefficient Extraction

• Extract constants (p, q, r) to solve specific forms.
• Steps: Plug values, solve equations, find auxiliary function coefficients.

Special Cases

• Functions of n: Deal with special forms like n², a^n, etc.
• General form: Use specified solutions as templates.

Examples and Problems

Example 1

• Recurrence Relation: an + 3a(n-1) + 2a(n-2) = n² + n + 1
• Solution Process:
  • Find auxiliary equation roots.
  • Solve for specific form p(n), q(n), and r(n).

Example 2

• Equation: a(n+1) + 5a(n) + 6a(n-1) = 2^n
• Steps:
  • Identify the form of p(n).
  • Solve auxiliary equation: x² + 5x + 6 = 0
  • Compare coefficients, find p, q.

Complex Roots Example

• Using Euler's formula: Apply polar coordinates, find magnitude and angle.
• Example Relation: a(n+1) + 4a(n) + 4a(n-1) = n3^n

Final Solutions

• Combination of Complementary Function (CF) and Particular Integral (PI).
• General Formula: CF + PI

Conclusion

• Recurrence relations connect terms in a sequence using relations to previous terms.
• Solving involves finding auxiliary equations and applying appropriate methods based on root types.
• Constant coefficients simplify the process, while special forms and complex numbers introduce variety.

Note: Practice solving provided problems to reinforce understanding. Follow provided methods step-by-step to achieve accuracy.