Coupled Time-Fractional Differential Equations
Some marriages survive. Some end in divorce. Love in a romantic relationship can die or stabilise as predictable behaviour as one grows old. Mathematical modelling of the phenomenon has been attempted again and again in the last three decades and the tools and techniques needed to deal with the complex phenomenon have been growing.
Manish Goyal and Shivangi Gupta from the GLA University, Mathura and Amit Prakash from the NIT Kurukshetra recently added new dimensions to the topic. Romantic love and interpersonal relationships are not continuous in time. Those furtive glances, the stolen kisses and sweet nothings are discrete in time. So differential equations that assume space and time continuity cannot be applied, the they reasoned. Therefore, the team used fractional differential equations.
This relatively new branch of calculus has found applications in many areas where mathematicians could not enter earlier. Besides being able to deal with discontinuities, it can deal with the memory of the loved one that colours present perceptions.
The team took a nonlinear system of coupled time-FDEs or fractional differential equations because Romeo’s love (or hate) for Juliet and Juliet’s love (or hate) for Romeo influence each other. Such coupled equations do not have easy solutions. They have to be solved by numerical methods.
The team used two alternative methods to solve the coupled fractional differential equations: fractional variation iteration where small changes in love (or hate) are iteratively calculated, and fractional homotopy perturbation transform, where small perturbations transform the ‘shape’ of the relationship. The researchers found that fractional variation iteration works better than fractional homotopy perturbation transform, and is easier to compute besides being powerful and reliable.
The team assumed that relationships decay exponentially when Romeo and Juliet are separated. The role of oxytocin and the notion of hearts growing fonder with separation are not taken into account. In spite of this limitation, the model will perhaps work not only in the analysis of romantic love (X) but also in other situations (Y, Z) where such coupled fractional differential equations can be applied. After all, once oranges and apples are converted to X and Y in mathematics, five plus two is seven whether we are counting oranges or apples.
PRAMANA-J. Phys, 92 (5): 82 (2019); DOI: 10.1007/s12043-019-1746-y
Udham P K, Freelance Science writer, Pune