JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III FINITE ELEMENT METHODS Previous Question Paper (March - 2021 - R18)

Approximate solutions are numerical or analytical techniques used to find a function that satisfies a differential equation and its boundary conditions with reasonable accuracy, even when an exact closed-form solution is impossible or computationally prohibitive. These methods typically involve assuming a trial function with unknown coefficients and minimizing an error functional.

The Raleigh-Ritz method is a variational technique used to find approximate solutions for boundary value problems by minimizing the total potential energy of the system. The steps are:

1. Assume an approximate solution in the form of a trial function, typically a linear combination of basis functions: Φ(x) = Φ₀(x) + Σ cᵢψᵢ(x), where Φ₀ satisfies the non-homogeneous boundary conditions and ψᵢ satisfy the homogeneous boundary conditions.
2. Formulate the total potential energy functional Π for the system.
3. Substitute the trial function into the functional Π.
4. Minimize the functional with respect to the unknown coefficients by setting the partial derivatives to zero: ∂Π/∂cᵢ = 0 for i = 1, 2, ..., n.
5. Solve the resulting system of linear algebraic equations for the coefficients cᵢ.

Given the differential equation (d²Φ/dx²) + Φ + x = 0 with Φ(0) = 0 and Φ(1) = 0. We assume an approximate solution of the form Φ(x) = c₁x(1-x) = c₁(x - x²), which satisfies the boundary conditions.

The residual R(x) is defined as:

Substituting Φ(x) = c₁(x - x²) and d²Φ/dx² = -2c₁:

According to the Galerkin method, the integral of the residual weighted by the basis function must be zero:

Evaluating the integrals:

Substituting back:

The approximate solution is Φ(x) = (5/18)(x - x²).