Fractional Differential Equations and Boundary Value Problems

Fractional differential equations extend classical calculus by allowing derivatives of non-integer order, thereby capturing memory and hereditary properties of diverse materials and processes. They encompass a spectrum of operators, among which the Riemann–Liouville and Caputo definitions are most prevalent. Boundary value problems in this context impose conditions at multiple points or in integral form, modelling situations such as viscoelastic beams, anomalous diffusion in porous media and control systems with after-effects. Analytical techniques hinge on fixed point theorems, spectral decompositions and integral transform methods, while numerical schemes employ finite differences, spectral approximations and multi-step approaches adapted to singular kernels. The interplay of global existence, uniqueness, stability and efficient approximation underpins the theory’s appeal. Applications range from thermal diffusion in complex geometries to biological population dynamics and electrochemical processes, all benefitting from the enhanced flexibility and fidelity offered by fractional dynamics.

Research from Nature Portfolio

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Research from all publishers

Research into fractional boundary value problems has advanced through novel formulations and solution techniques. A study of a p-Laplacian nonperiodic boundary value problem formulated with generalised Caputo derivatives established existence and uniqueness of solutions via Banach and Schauder fixed point theorems, illustrating the approach with a concrete example that emphasises the method’s adaptability to nonlinear spatial operators. Another investigation extended a classical thermostat differential equation to a hybrid fractional model under mixed boundary conditions. By exploiting Dhage’s fixed point results for single-valued and set-valued mappings, the authors proved solution existence and provided illustrative simulations, highlighting potential applications in thermal control systems with after-memory effects. A further contribution analysed a coupled system of fractional differential equations tied by integral boundary conditions. Employing increasing φ-concave operator theory, this work derived conditions ensuring unique solvability and demonstrated the results with a practical example, thereby enriching the repertoire of solvable coupled fractional systems encountered in engineering and physics.

Fractional Differential Equations and Boundary Value Problems publication trend

The graph below shows the total number of articles in fractional differential equations and boundary value problems across all publications each year (not limited to Nature Index journals).

Technical terms

Fractional derivative: A generalisation of the classical derivative to non-integer orders, capturing history-dependent dynamics through integral definitions.

Boundary value problem: A differential equation supplemented by conditions specified at two or more points or in an integral form, dictating global solution behaviour.

Caputo derivative: A fractional derivative defined via an integral operator that allows for initial conditions expressed in terms of integer-order derivatives.

Riemann–Liouville derivative: A foundational fractional derivative defined by an integral transform that generalises the nth-order differentiation operator.

p-Laplacian operator: A nonlinear differential operator extending the Laplacian by incorporating a variable exponent p, modelling non-Newtonian diffusion processes.

References

1. Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives. Advances in Continuous and Discrete Models (2021).
2. A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Boundary Value Problems (2020).
3. Unique solutions for a new coupled system of fractional differential equations. Advances in Continuous and Discrete Models (2018).