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In this work, the modified simple equation (MSE) method is used to find exact traveling wave solutions to nonlinear evolution equations (NLEEs) in mathematical physics. To do so we have used the nonlinear Klein–Gordon (nKG) equation and the (2 + 1)-dimensional modified Zakharov–Kuznetsov (mZK) equation. Two classes of exact explicit solutions-hyperbolic and trigonometric solutions of the associated NLEEs are characterized with some free parameters. It is shown that the method provides a powerful mathematical tool for solving NLEEs in mathematical physics and engineering fields.