For each adaptive approximation, the collocation points are selected on the computational interval using a uniform nodal rule. Let the computational interval be t ∈ [ a , b ] , and let N c denote the number of collocation points. The nodes are defined by

At every adaptive iteration, the governing differential equation is enforced at the same set of N c collocation points. Thus, for an m -th order differential equation, the differential-equation residual equations are evaluated at N c points, while the prescribed initial or boundary conditions are imposed separately.

If N A = | A | denotes the number of active Fibonacci modes, the resulting nonlinear system is assembled from the collocation residual equations and the required initial/boundary constraint equations. In particular, for an m -th order problem with m independent conditions, the system contains N c + m equations before any equation replacement or row selection is applied.

To obtain a square nonlinear algebraic system when required, m collocation equations are replaced by the corresponding initial or boundary conditions. The replacement is performed at the rows associated with the selected constraint locations. Consequently, the final system contains N c equations for N A unknown Fibonacci coefficients when N A = N c .

For first-order initial-value problems, one collocation equation is therefore replaced by the initial condition. For higher-order initial-value or boundary-value problems, the number of replaced equations is equal to the number of independent prescribed conditions.

The same node construction and equation-assembly rule is retained throughout the enrichment and pruning stages so that changes in the numerical solution can be attributed to the active Fibonacci basis rather than to a changing collocation grid.

Because the adaptive procedure may produce a nonconsecutive active Fibonacci index set, the differentiation operator cannot in general be defined by simply restricting the full Fibonacci differentiation matrix to the active indices. Let the candidate basis be

and let the corresponding full differentiation matrix be D , defined by

For an active index set

the approximation is written as

where Φ A contains only the active Fibonacci modes. The derivative is nevertheless evaluated through the full candidate-space differentiation operator:

where D C , A denotes the columns of the full differentiation matrix corresponding to the active modes.

Thus, if the derivative of an active mode F j k ( t ) contains Fibonacci modes whose indices are not members of A , those modes are not discarded. They remain in the derivative representation within the candidate space and are included when the differential-equation residual is evaluated.

More explicitly,

Consequently,

The same construction is applied recursively for higher-order derivatives. For example,

where the active columns are selected only after the full candidate-space differentiation operator has been constructed.

This formulation separates basis selection from differentiation. The adaptive procedure determines which Fibonacci modes are used in the approximation itself, whereas differentiation is performed in the full candidate space. Therefore, nonconsecutive active sets do not introduce an artificial truncation of derivative contributions. A mode is removed from the approximation only through the prescribed pruning criterion, and not merely because its derivative representation is outside the active set.

At each adaptive iteration, the nonlinear algebraic system associated with the current active Fibonacci set is solved for the unknown expansion coefficients using the MATLAB fsolve solver. For the initial active set, the coefficient vector is initialized consistently with the prescribed initial or boundary conditions. At subsequent stages, the converged coefficient vector obtained from the preceding iteration serves as an effective initial guess for the current solve. The iterative solution process is terminated when both the nonlinear-system residual and the coefficient update satisfy a numerical tolerance of ε solver = 10 − 10 .

The adaptive residual tolerance and the coefficient-pruning tolerance are denoted by τ res and τ prune , respectively. Following a successful nonlinear solve, the differential residual norm is evaluated to govern the adaptive enrichment and pruning criteria. If a proposed basis modification leads to a failure in fsolve convergence within the prescribed iteration limit, the tentative modification is rejected, and the previous converged active basis and coefficient vector are strictly retained. The procedure then either investigates the next admissible basis adjustment or terminates if convergence criteria are met, ensuring a consistent and robust numerical implementation across all benchmark problems.

Algorithm 1. Adaptive Sparse Fibonacci Collocation Method

Input: Candidate Fibonacci index set C , initial active set A ( 0 ) , collocation nodes, initial/boundary conditions, nonlinear-solver tolerance ε N L , residual tolerance ε R , coefficient-pruning tolerance ε c , and maximum number of adaptive iterations K max .

Output: Final active index set A ∗ and coefficient vector c ∗ .

1. Initialize A ← A ( 0 ) and construct the corresponding Fibonacci approximation.

2. Assemble the nonlinear collocation system using the current active set and the prescribed initial/boundary-conditions.

3. Initialize the coefficient vector. For the first iteration, use the prescribed conditions to determine the constrained coefficients and initialize the remaining coefficients. At subsequent iterations, use the previously converged coefficient vector as the initial guess.

4. Solve the nonlinear algebraic system using the selected nonlinear solver until the prescribed nonlinear-solver tolerance ε N L is satisfied.

5. If the nonlinear solve is unsuccessful, reject the current tentative basis modification and retain the previous converged active set and coefficient vector.

6. Evaluate the collocation residual

7. Apply the coefficient-based pruning criterion and remove a mode only when its removal satisfies the prescribed pruning condition.

8. For each inactive candidate mode, evaluate the residual reduction obtained by tentative enrichment.

9. If several candidate modes satisfy the enrichment criterion, select the mode giving the largest admissible residual reduction.

10. Add the selected mode, reassemble the nonlinear system, and re-solve for all active coefficients.

11. If the nonlinear solve associated with the enrichment is unsuccessful, reject that enrichment and retain the previous converged representation.

12. Repeat the enrichment-pruning cycle until

no admissible enrichment produces a meaningful residual reduction, and no active coefficient satisfies the pruning criterion.

13. If K max is reached, terminate the adaptive procedure and retain the latest converged representation.

14. Return A ∗ and c ∗ .

Consider the nonlinear differential equation

subject to the initial conditions

The exact solution is

For an active Fibonacci index set A , the ASFC approximation is written as

The initial conditions are imposed directly on the approximation,

The remaining coefficients are determined from the nonlinear collocation residual

evaluated at the prescribed collocation points. The adaptive procedure introduces additional Fibonacci modes when the residual reduction obtained from an inactive candidate mode satisfies the enrichment criterion. Conversely, insignificant active modes are removed when their elimination does not produce a significant deterioration of the residual.

This problem is particularly useful for examining sparse basis selection because the exact solution contains a constant component and an oscillatory component while the governing equation includes quadratic nonlinearities.

The second benchmark is a quadratic Riccati equation defined on

by

with the initial condition

and exact solution

Equivalently, the equation can be written in the standard quadratic Riccati form

where

Using the ASFC representation, the exact nonlinear differential equation is transformed into a finite-dimensional nonlinear collocation system. The coefficient associated with the initial condition is constrained through Equation (19), while the remaining coefficients are determined from the collocation equations generated by the residual of Equation (18).

Because the exponential exact solution contains both even and odd powers in its Taylor expansion, this benchmark provides a useful test of adaptive Fibonacci basis enrichment and the ability of the proposed method to construct a compact representation without imposing a fixed complete truncation.

The third benchmark considers the nonlinear third-order differential equation

subject to

The exact solution is

Starting from the active Fibonacci representation,

the required derivatives are evaluated using the Fibonacci differentiation operator described in Section 4. The three initial conditions are imposed through

The remaining coefficients are obtained from the nonlinear collocation residual

Additional Fibonacci modes are introduced when they provide a meaningful reduction of the residual, whereas insignificant modes are removed according to the pruning criterion. The process continues until the prescribed residual criterion is satisfied or no admissible enrichment provides a sufficient improvement.

This problem also provides a test of the proposed method for a higher-order nonlinear equation and explicitly verifies the treatment of the second derivative initial condition u ″ ( 0 ) = − 1 .

Before computing the numerical errors, the exact solutions used in the three benchmark problems are independently verified against their corresponding governing differential equations and initial conditions. The verification is performed symbolically and is independent of the ASFC approximation. For each problem, the exact-solution verification residual is defined by substituting the exact solution into the corresponding differential equation.

6.4.1. Verification of Problem 1

For Problem 1,

and therefore

Substitution into the governing equation gives

The initial conditions are also satisfied exactly:

Consequently,

6.4.2. Verification of Problem 2

For Problem 2,

Substitution into the governing Riccati equation yields

The prescribed initial condition is verified by

Hence,

6.4.3. Verification of Problem 3

For Problem 3,

with derivatives

Furthermore,

Therefore,

The initial conditions are satisfied exactly:

Thus,

The symbolic verification results are summarized in Table 1.

Table 1.Symbolic verification of the exact solutions for the three benchmark problems.

These results confirm that the three exact solutions satisfy their respective governing equations and prescribed initial conditions identically. The subsequent numerical error calculations are therefore performed relative to independently verified reference solutions.

Let u N ( t ) denote the ASFC approximation and u ex ( t ) the corresponding exact solution. The pointwise absolute error is defined by

The maximum absolute error over the computational interval is

For numerical evaluation, the maximum is calculated on the prescribed error-evaluation grid { t q } q = 1 N e ,

In addition to the solution error, the maximum absolute differential-equation residual is used to measure the consistency of the numerical approximation with the governing equation. For a general differential operator ℱ ,

and

On the numerical evaluation grid, this becomes

For the three benchmark problems, the corresponding residuals are

and

Accordingly, the problem-dependent maximum residual is defined as

The sparsity of the final Fibonacci representation is quantified by the number of retained active modes,

where A is the final active Fibonacci index set. The corresponding compression or sparsity ratio is defined by

where N cand denotes the number of candidate Fibonacci modes considered by the adaptive procedure. A larger value of S corresponds to a more compact final representation.

Thus, the numerical assessment is based on three complementary measures: E ∞ , R ∞ and N act (or S ). The first measures the deviation from the independently verified exact solution, the second measures satisfaction of the governing differential equation, and the third quantifies the compactness of the adaptive Fibonacci representation.

Figures 1(a)-(c) provide graphical comparisons of the ASFC results for the three benchmark problems with respect to the independent variable t ∈ [ 0 , 1 ] . For each individual problem, the numerical approximation u ( t ) , pointwise absolute error | e ( t ) | , and differential-equation residual | R ( t ) | are systematically displayed alongside the Bernoulli polynomial results [1]. These subfigures serve as an explicit visual complement to the quantitative error and residual measures defined above, while the detailed discussion and interpretation of the numerical performance are presented in Section 7.

(a)

(b)

(c)

Figure 1. (a) Numerical solution, pointwise absolute error, and differential-equation residual for Problem 1 (Nonlinear Oscillator) obtained by the proposed Adaptive Sparse Fibonacci Collocation (ASFC) method and the Bernoulli polynomial method [1]; (b) Numerical solution, pointwise absolute error, and differential-equation residual for Problem 2 (Riccati Equation) obtained by the proposed Adaptive Sparse Fibonacci Collocation (ASFC) method and the Bernoulli polynomial method [1]; (c) Numerical solution, pointwise absolute error, and differential-equation residual for Problem 3 (Third-Order Nonlinear Equation) obtained by the proposed Adaptive Sparse Fibonacci Collocation (ASFC) method and the Bernoullipoly nomial method [1].

The main numerical characteristics of the three benchmark problems are summarized in Table 2.

Table 2. Numerical performance of the proposed ASFC method.

The results show that the adaptive procedure does not retain the complete candidate Fibonacci basis. For Problem 1, only seven modes are retained, whereas eight modes are sufficient for Problem 2 and ten modes are retained for Problem 3. This behavior illustrates the principal sparse-approximation feature of the proposed formulation.

For Problem 2, the retention of eight modes from a 25-mode candidate basis corresponds to a compression ratio of 68.0%. Thus, approximately one-third of the candidate modes are retained in the final representation. The result is particularly relevant because the exact solution is exponential rather than polynomial, providing a useful test of the ability of the Fibonacci basis to represent smooth non-polynomial behavior.

Figures 1(a)-(c) compares the numerical and exact solutions for all three benchmark problems together with the corresponding pointwise absolute errors and differential-equation residuals.

For Problem 1, the computed solution reproduces the oscillatory behavior of the exact solution over the complete interval. The use of seven active Fibonacci modes demonstrates that the oscillatory solution can be represented without retaining the entire candidate basis.

Problem 2 provides a different test because its exact solution is u ( t ) = e t . Despite the exponential character of the solution and the quadratic Riccati nonlinearity, the adaptive procedure retains only eight modes. The resulting approximation follows the exact solution throughout the computational interval, while the pointwise error remains controlled.

Problem 3 represents the most demanding benchmark in terms of differential order, since the governing equation contains a third derivative and a nonlinear exponential term. The exact solution u ( t ) = ln ( 1 + t ) is also non-polynomial. The final approximation uses ten active Fibonacci modes and reproduces the qualitative behavior of the reference solution over 0 ≤ t ≤ 1 .

The maximum absolute errors reported in Table 2 provide the quantitative counterpart of these observations. In contrast to a visual comparison alone, E ∞ measures the largest deviation between the numerical and independently verified exact solutions over the prescribed computational grid.

Solution error alone is not sufficient to characterize the performance of a nonlinear collocation method. Therefore, the differential-equation residual is evaluated independently using

The corresponding maximum residual R ∞ measures the extent to which the computed approximation satisfies the governing differential equation.

The residual curves depicted in Figure 1(a3), Figure 1(b3), and Figure 1(c3) remain strictly bounded across the computational domain t ∈ [ 0 , 1 ] for all three benchmark problems. In each case, the residual magnitude | R ( t ) | associated with the ASFC method remains consistently suppressed below 10⁻⁷, demonstrating substantial consistency with the governing differential equations compared to the Bernoulli polynomial approach [1]. This behavior is directly aligned with the residual-controlled adaptive strategy introduced in Section 4, wherein inactive Fibonacci modes are selectively incorporated into the active basis only when they provide an admissible reduction in the differential residual.

The residual results are particularly important for Problem 3. Because the equation is third-order and contains the stiff nonlinear exponential term − 2 e − 3 u ( t ) , subsequent differentiation and nonlinear evaluations may significantly amplify numerical approximation errors. As demonstrated in Figure 1(c3), the residual | R ( t ) | remains bounded on the order of 10⁻⁸, thereby providing an independent confirmation that the obtained sparse Fibonacci approximation rigorously satisfies the original governing equation rather than merely reproducing the qualitative shape of the exact solution.

The numerical results also illustrate the role of adaptive basis selection. The final approximation is written as

where A is the adaptively determined active index set.

The number of retained modes is seven, eight, and ten for Problems 1-3, respectively. These values indicate that the required approximation dimension depends on the structure and differential order of the problem rather than being fixed uniformly in advance.

The compression ratio provides an additional measure of this behavior. In particular, the 68.0% compression obtained for Problem 2 demonstrates that the candidate space can be substantially reduced while maintaining the required numerical accuracy. The same interpretation applies to Problems 1 and 3, for which the same 25-mode candidate space is reduced to 7 and 10 active modes, corresponding to compression ratios of 72.0% and 60.0% respectively (Table 2).

An important consequence is that increasing the candidate basis does not automatically imply that all additional modes must be retained. Instead, the adaptive procedure attempts to identify modes that contribute meaningfully to residual reduction. This distinguishes the proposed ASFC formulation from conventional fixed-order Fibonacci collocation approaches in which the complete truncated basis is retained.

Figures 1(a)-(c) also provides a comparison with the Bernoulli polynomial results reported by Bicer and Sezer [1]. The comparison includes the numerical solution, pointwise absolute error, and differential-equation residual for the benchmark problems.

The comparison should be interpreted using the reported numerical quantities rather than the solution curves alone. In particular, E ∞ characterizes approximation accuracy relative to the exact solution, while R ∞ measures consistency with the governing differential equation. The number of active Fibonacci modes additionally characterizes the compactness of the proposed representation.

The ASFC formulation differs from the Bernoulli approach primarily in the construction of the approximation space. The Bernoulli formulation uses a prescribed polynomial representation, whereas the present method introduces a residual-controlled mechanism for selecting an active subset of Fibonacci modes. Consequently, the comparison provides information not only about approximation accuracy but also about the effect of adaptive basis selection.

Taken together, the three experiments cover oscillatory, exponential, and logarithmic solution behaviors and include first- and third-order nonlinear differential operators. The numerical results indicate that the proposed ASFC framework can represent these solutions using relatively small active Fibonacci spaces.

Three observations are particularly relevant. First, the numerical solutions remain consistent with the independently verified exact solutions. Second, the differential-equation residual provides an additional measure of numerical consistency and forms the basis of the adaptive enrichment mechanism. Third, the final active basis contains fewer modes than the corresponding candidate space, demonstrating the sparse character of the proposed approximation.

The results therefore support the central computational concept of the method: rather than prescribing a fixed approximation order and retaining every available basis function, the ASFC procedure seeks an active Fibonacci representation adapted to the nonlinear residual structure of the problem.

A complete assessment of computational efficiency would additionally require reporting implementation-dependent quantities such as CPU time, nonlinear-solver iterations, conditioning measures, and the number of enrichment/pruning iterations. These quantities are not introduced here unless they are directly available from the numerical implementation.