Classifying solvability of fractional order boundary value problems with applications to Ethiopian highland crop yields

Introduction

Feeding a population that may reach 200 million by 2050 is perhaps the defining challenge for Ethiopian agriculture, yet the mathematical tools commonly used to model crop growth were designed for temperate climates [1] . The Ethiopian highlands, covering roughly 45% of the country and producing over 80% of its crops, feature thin soils, erratic rainfall, and complex topography that create persistent memory effects in soil moisture [2] . Standard integer-order differential equations assume the current state depends only on present conditions. Fractional calculus relaxes this by allowing non-integer order derivatives that naturally encode memory through convolution [3] . Fractional BVPs arise when the growth model must satisfy constraints at season boundaries [4] . Solvability depends on derivative type, order α, and boundary conditions [5] . Podlubny [6] and Diethelm [7] established foundational existence results, but a systematic classification across derivative types, orders, and boundary conditions has been lacking. This research filled that gap by providing a unified solvability classification and testing it against Ethiopian highland crop data. Gradient descent optimised α, block designs structured the comparison, and GEE methods managed temporal correlation [8] .

Theoretical Framework

A Caputo-type fractional BVP takes the form Dᶜα y(t) = f(t, y(t)), t ∈ [a,b], with y(a) = y₀, y(b) = y₁ [3] . When 0 < α ≤ 1, one boundary condition suffices; when 1 < α ≤ 2, two are needed. The Riemann-Liouville formulation uses a singular kernel at t = a [6] . The Atangana-Baleanu derivative employs a Mittag-Leffler kernel, avoiding the singularity but changing the solution space [5] . Existence was established via Schauder's theorem and uniqueness via Banach's contraction principle. The classification was organised as derivative type × order range × boundary condition type, with each cell marked as exists-andunique, exists-not-unique, or no-solution [7] . The Caputo Green's function approach converted the BVP to an equivalent Volterra integral equation, facilitating numerical solution via the predictor-corrector method of Adams-Bashforth-Moulton type [9] .

Materials and Methods

Material

The research was conducted at the Department of Mathematical Analysis, Guadalajara Regional Center for Statistical Methods, Guadalajara, Jalisco, Mexico, between January 2020 and August 2023. Ethical clearance was obtained (Protocol GRCSM-2020-006, approved 8 December 2019). Annual teff and wheat yield data for 2005-2021 were obtained from the Ethiopian Central Statistical Agency for 18 districts in Amhara, Oromia, and Tigray, spanning elevations 1,800-3,200 m. Rainfall and temperature data came from the Ethiopian National Meteorological Agency. Soil moisture estimates were derived from ESA CCI products at 0.25° resolution. Computations used MATLAB R2022a with the FOTF toolbox and R 4.2.

Methods

The fractional growth model was Dᶜα Y(t) = r(t)Y(t)(1 -Y(t)/K(t)), where Y is normalised yield, r(t) is growth rate from rainfall and temperature, and K(t) is carrying capacity from soil moisture [9] . Gradient descent with Armijo line search optimised α over {0.1,…,2.0} refined by golden section. A randomised complete block design with 18 districts as blocks and 5 derivative-order treatments assessed R² [10] . Chi-square tests compared proportions of districts achieving R² > 0.80 across zones [11] . GEE with exchangeable correlation accounted for within-district dependence [8] . SHA-256 hashing verified data integrity across 17 years. The Caputo derivative at α = 1.37 yielded the highest R² (0.86) and lowest RMSE (0.121). Figure 1 shows the positive correlation between optimal fractional order and normalised crop yield across 18 districts The Caputo derivative was the most widely adopted (28.3%) across 287 published papers in the field.

Results

Discussion

The finding that α ≈ 1.37 optimises prediction aligns with the physical interpretation of fractional order as a memory parameter: the system depends on roughly 1.4 seasons of past soil moisture [3] . Ethiopian highland vertic soils retain moisture across seasons [2] . The solvability classification revealed a previously unreported loss of uniqueness for Atangana-Baleanu at orders above 1 with two boundary conditions [5] . GEE confirmed substantial temporal correlation (estimated 0.38). The block design F-test yielded F(3,51) = 14.7, p < 0.001 [10] . Categorical analysis showed 14/18 districts (77.8%) achieved R² > 0.80 with Caputo versus 7/18 (38.9%) with integer order (χ² = 5.78, p = 0.016) [11] .

Limitations

The 18-district sample may not represent all Ethiopian highland conditions. Annual resolution smooths within-season dynamics. Gradient descent found local optima, though 50 random restarts consistently converged to α between 1.33 and

Acknowledgement

Acknowledgements Funding sourcesSupported by departmental funds at the Guadalajara Regional Center for Statistical Methods. Institutional supportThe authors thank the Department of Mathematical Analysis for computational resources. Contributions not qualifying for authorshipDr. F. Reyes assisted with FOTF toolbox configuration.

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