Question

Since b, a, k are fixed positive real numbers, consider the equation of heat with prewritten boundary conditions: ut(x, t) = kuzz(x, t) – a u(x,t) 0 < x <b, t> 0, u(x,0) = f(x), u(0,t) = 0 u(b, t) = 0, Assuming that u (x, t) = v(x)w(t), solve the problem of self-value for the system of ordinary differential equations obtained by means of separation of variables. Then, express the solution of the equation as a Fourier series, and find the Fourier coefficient. Finally, if f(x) = M0, find the explicit solution in Fourier series

please can the answer can be visible to read and explain each step

Thanks

Answer 1

Solchon o Since b consider the equation Conditions. ak are ne equation freed poniture of heat with real numbers pre-wistten de boundary Olli,+) = Koy =kuxxlxt)-2u(xit) ) olxeb, to veo) = f(x), UCO,t =0 Ub,t) = 0 The heat equation which can also be written as. ou = kane - ot Now Ult, ta valit! ve) wat) = ku"(x) wit) - and RHS is independent of Now LH-S is independent at So, we can writh Vla - kwit) lit) WIEVGK X 34 (e) = 1 { wilt at a Wit) vaw (t)) wt val so and to e nter way.wce) -- ☆ VIC)-oven) 20 1 0 wilt) + x Wiltta -ko WE=6 V( )

The of eyno vex)= general solution x + cesor kovce) kovce, & pekot of ano, vrees wilt) + 1 Z=0 The general solution in vx= dit dex wit) 7 . + +d3 vo weest) (d, or edzetorget to etork) kov) ac and cy are orbetrang for oto and constants. &a, t) = diadze & t+d3l where di-ci, dz=ch, d3=63 Substituting the soundary en we have 6= aklat Jo = kaq and dz=-d vex) w (t)=cle son box) (& 3G sinka, ). ( kov) & Gekort) tez ketat ) klava) We know that, klat 2 klavea) e, t) = & a sinkona (te te gee ) f(x) 2018, 0) = { casin (ka x) ( at vent (3) where ag = 5 fla) sin kar de and lab as given in ka Conditions.