This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations.

In this paper, we establish several new $(p,q)$-integral identities involving $(p,q)$-integrals by using the definition of a $(p,q)$-derivative. These results are then used to derive $(p,q)$-integral Simpson- and Newton-type inequalities involving convex functions. Moreover, some examples are given to illustrate the investigated results.

In this paper, we prove a new quantum integral equality involving a parameter, left and right quantum derivatives. Then, we use the newly established equality and prove some new estimates of quantum Ostrowski, quantum midpoint, quantum trapezoidal and quantum Simpson’s type inequalities for q-differentiable convex functions. It is also shown that the newly established inequalities are the refinements of the existing inequalities inside the literature. Finally, some examples and applications are given to illustrate the investigated results.

In this paper, we establish a new ${(p,q)}^b$-integral identity involving the first-order ${(p,q)}^b$-derivative. Then, we use this result to prove some new ${(p,q)}^b$-integral inequalities related to Hermite–Hadamard inequalities for ${(p,q)}^b$-differentiable convex functions. Furthermore, our main results are used to study some special cases of various integral inequalities. The newly presented results are proven to be generalizations of some integral inequalities of already published results. Finally, some examples are given to illustrate the investigated results.

This study proves equality for differentiable functions involving the conformable fractional integrals. Using the established identity, we offer new Simpson type inequalities for convex functions via conformable fractional integrals. We also consider some special cases which can be deduced from the main results.