Integral transformations have been used for a long time in the solution of differential equations either solely or combined with other methods. These transforms provide a great advantage in reaching solutions in an easy way by transforming many seemingly complex problems into a more understandable format. In this study, we used an integral transform, namely Kashuri Fundo transform, by blending with the homotopy perturbation method for the solution of non-linear fractional porous media equation and time-fractional heat transfer equation with cubic non-linearity.

Recently, it has become quite common to investigate the solutions ofproblems that have an important place in scientific fields by using integraltransforms. The most important reason for this is that this transform allows thesimplest and least number of calculations to be made while reaching the solutionsof the problems. In this study, we are looking for a solution to the decay problem,which has a very important place in fields such as economics, chemistry, zoology,biology and physics, by using the Kashuri Fundo transform, which is one of theintegral transforms. In order to reveal the ease of use of this transform in reachingthe solution, some numerical applications were examined. The results of thesenumerical applications reveal that the Kashuri Fundo transform is quite efficient inreaching the solution of the decay problem.

Differential equations are expressions that are frequently encountered in mathematical modeling of laws or problems in many different fields of science. It can find its place in many fields such as applied mathematics, physics, chemistry, finance, economics, engineering, etc. They make them more understandable and easier to interpret, by modeling laws or problems mathematically. Therefore, solutions of differential equations are very important. Many methods have been developed that can be used to reach solutions of differential equations. One of these methods is integral transforms. Studies have shown that the use of integral transforms in the solutions of differential equations is a very effective method to reach solutions. In this study, we are looking for a solution to damped and undamped simple harmonic oscillations modeled by linear ordinary differential equations by using Kashuri Fundo transform, which is one of the integral transforms. From the solutions, it can be concluded that the Kashuri Fundo transform is an effective method for reaching the solutions of ordinary differential equations.

Integral equations can be defined as equations in which unknown function to be determined appears under the integral sign. These equations have been used in many problems occurring in different fields due to the connection they establish with differential equations. Abel?s integral equation is an important singular integral equation and Abel found this equation from a problem of mechanics, namely the tautochrone problem. This equation and some variants of it found applications in heat transfer between solids and gases under non-linear boundary conditions, theory of superfluidity, subsolutions of a non-linear diffusion problem, propagation of shock-waves in gas field tubes, microscopy, seismology, radio astronomy, satellite photometry of airglows, electron emission, atomic scattering, radar ranging, optical fiber evaluation, X-ray radiography, flame and plasma diagnostics. Integral transforms are widely used mathematical techniques for solving advanced problems of applied sciences. One of these transforms is the Kashuri Fundo transform. This transform was derived by Kashuri and Fundo to facilitate the solution processes of ODE and PDE. In some works, it has been seen that it provides great convenience in finding the unknown function in integral equations. In this work, our aim is to solve Abel?s integral equation by Kashuri Fundo transform and some applications are made to explain the solution procedure of Abel?s integral equation by Kashuri Fundo transform.

Integral transforms provide us great convenience in finding exact and approximate solutions of many mathematical physics and engineering problems such as signals, wave equation, heat conduction, heat transfer. In this study, we consider the Kashuri Fundo transform, which is one of these integral transforms, and our aim is to show that this transform is an effective method in solving steady heat transfer problems and obtained results are compared with the results of the existing techniques.