Remarks on Sequential Caputo Fractional Differential Equations with Fractional Initial and Boundary Conditions

In the literature so far, for Caputo fractional boundary value problems of order 2 q when 1 < 2 q < 2 , the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order 2 q . Further, even the initial conditions for a Caputo fractional differential equation of order n q use the corresponding integer-order initial conditions of order ‘ n ’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘ n q ’ will only be a C n q solution and not a C n solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order k q , k = 0 , 1 , … ( n − 1 ) when the order of the fractional derivative involved in the differential equation is n q . The Caputo left derivative can be computed only when the function can be expressed as f ( x − a ) . Then the Caputo right derivative of the same function will be computed for the function f ( b − x ) . Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when q = 1 . Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data.