This study examines a new approach for the approximate solution of hyperbolic telegraph equations emerging in magnetic fields and electrical impulse transmissions. We introduce a Laplace-Carson transform coupled with the homotopy perturbation method which is called the Laplace-Carson homotopy perturbation method ( L c -HPM). The most significant feature of this approach is that we do not require any restriction of variables and hypotheses to find the results of nonlinear problems. Further, HPM using He’s is applied to reduce the number of computations in nonlinear terms. We demonstrate some graphical results to show that L c -HPM is a simple and suitable approach for linear and nonlinear problems.

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The sign-invariant theory is used to study the radially symmetric nonlinear diffusion equations with gradient-dependent diffusivities. The first-order non-stationary sign-invariants and the first-order non-autonomous sign-invariants admitted by the governing equations are identified. As a consequence, the exact solutions to the resulting equations are constructed due to the corresponding reductions. The phenomena of blow-up, extinction and behavior of some solutions are also described.