Theoretical analysis on the nonlinear fractional differential equations and generalized heat equation

Using Schauder's fixed-point theorem, we establish sufficient conditions for the existence and uniqueness of solutions to the nonlinear fractional boundary value problem: {cases} {}cDβζ(x) + f(x, ζ(x), Iγζ(x)) = 0, & x ∈ I = [0, 1], 1 0, \\?(0) = 0, ζ(1) = φ(ζ), {cases} {(0.1)} where φ is a functional defined on C(I, R) . By constructing an appropriate Green''s function, we derive a Lyapunov-type inequality for a special case of the problem (0.1): {cases} {}cDβζ(x) + λ(x)Iγζ(x) = η(x, ζ(x)), & x ∈ I = [0, 1], 1 0, \\?(0) = 0, ζ(1) = φ(ζ). {cases} {(0.2)} We further make an analysis for equation (0.2) by applying the inverse operator method and the Mittag-Leffler function with illustrative examples demonstrating applications obtained. Finally, we construct an analytic solution to the following generalized fractional heat equation with an initial condition in n dimensions based on an inverse operator: {cases} {}cDₜαu(t, x) = Δ_{a₁(x₁), ⋯, aₙ(xₙ)}u(t, x) + f(t, x), & (t, x) ∈ R⁺ × Rⁿ, 0 < α ≤ 1, \(0, x) = ψ(x), {cases} {(0.3)} where Δ_{a₁(x₁), ⋯, aₙ(xₙ)} = a₁(x₁){∂²}{∂ x₁²} + ⋯ + aₙ(xₙ){∂²}{∂ xₙ²}.